Modelos_Estatisticos-2024-08-10
Helena R. S. D’Espindula
2024-08-10
Regressão Linear
### Modelos estatísticos - Aula de modelos lineares
### Pacotes usados na aula.
require("ISLR")## Carregando pacotes exigidos: ISLRrequire("ggplot2")## Carregando pacotes exigidos: ggplot2require("GGally")## Carregando pacotes exigidos: GGally## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2require("leaps")## Carregando pacotes exigidos: leapsrequire("car")## Carregando pacotes exigidos: car## Carregando pacotes exigidos: carDataoptions(device = X11)Recomendações de leitura
[…]
Objetivo
Num modelo de regressão linear, a relação entre uma variável aleatória de interesse (variável resposta) e um conjunto de preditores (variáveis explicativas) é definida por uma função linear envolvendo os preditores e um conjunto de parâmetros associados (coeficientes ou $$′s ou parametros).
“Composição”
- Variaveis Resposta
- Variaveis Explicativas
Exemplo:
- Aplicação 1: Preço de Imóveis
- Variável Resposta: Preço do imóvel
- Variáveis Explicativas: Tamanho do imóvel (m2), número de quartos, localização (bairro), idade do imóvel, presença de garagem
- Aplicação 2: Desempenho Escolar
- Variável Resposta: Nota final em uma disciplina
- Variáveis Explicativas: Número de horas estudadas por semana, frequência nas aulas, nota em provas anteriores, participação em atividades extraclasse, apoio familar
- Aplicação 3: Vendas de um produto
- Variável Resposta: Vendas mensais de um produto
- Variáveis Explicativas: Preço do produto, gastos em marketing, número de vendedores, promoções realizadas, condições econômicas.
- Aplicação 4: Consumo de Energia
- Variável Resposta: Consumo de energia elétrica (kWh)
- Variáveis Explicativas: Tamanho da residência (m2), número de moradores, uso de eletrodomésticos, temperatura média na região, tipo de aquecimento
- Aplicação 5: Rendimento de Investimentos
- Variável Resposta: Rendimento de uma carteira de investimentos
- Variáveis Explicativas: Taxa de juros, porcentagem de ações vs. renda fixa, valor inicial investido, taxa de inflação, duração do investimento
Modelo de regressão linear
Seja \(y\) a variável resposta e \(x1, x2, ..., xk\) os \(k\) preditores. O modelo de regressão linear fica definido por: \[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_k x_k + \epsilon \]
onde:
- \(\beta_0\) é o intercepto do modelo (se todos os \(x\)s e \(\epsilon\) forem 0 $ y = _0$ );
- \(\beta_1, \beta_2, ... \beta_k\) são parâmetroa do modelo, que correspondem aos efeitos lineares de cada um dos \(k\) preditores;
- \(\epsilon\) representa os erros aleatórios, específicos para cada indivíduo. É a parte de \(y\) que não é explicada pela regressão linear.
Erros
Embora sejam não observáveis e não explicados pelo modelo, na regressão linear assumimos as seguintes propriedades para os erros:
- Os erros têm média zero;
- Os erros têm variância constante;
- Os erros para dois indivíduos quaisquer são não correlacionados;
- Os erros têm distribuição normal.
Este conjunto de suposições é usualmente denotado por \(\epsilon \sim N (0, \sigma^2)\).
Regressão Linear Simples
Como caso particular da regressão linear temos a regressão linear simples, que se caracteriza por considerar um único preditor:
\[ y = \beta_0 + \beta_1 x + \epsilon, \space \space \space \space \space \epsilon \sim N (0, \sigma^2) \]
Regressão Linear Polinomial
Outro modelo de regressão linear bastante usual é o modelo de regressão polinomial, que permite explicar uma relação não linear entre a resposta e o(s) preditor(es):
\[
y = \beta_0 + \beta_1 x + \beta_2 x^2 + ... + \beta_k x^k + \epsilon,
\space \space \space \space \space
\epsilon \sim N (0, \sigma^2)
\] 
Fig 02
Fig 03
Fig 04
Fig 05
Fig 06
Um breve histórico
- Século 19: Desenvolvimento da teoria de mínimos quadrados, que serve de base para o ajuste de modelos de regressão.
- A teoria de mínimos quadrados teve origem na Física motivada, dentre outros, por problemas na área navegação (século 18).
- Ao longo do século 19, modelos lineares e o método de mínimos quadrados passaram a ser utilizados em outras ciências, baseados em modelos pré-estabelecidos, ou apenas em evidências empíricas.
- O termo regressão foi introduzido por Francis Galton em 1875, baseado no princípio da “regressão à média”.
Regressão linear
- Os dois objetivos principais de uma análise de regressão são os
seguintes:
- Exploratório- Analisar a relação funcional entre uma variável aleatória (resposta) e um conjunto de variáveis preditoras.
- Preditivo- Predizer valores não observados da variável resposta para valores especificados das variáveis preditoras.
Exemplo- Consumo de combustível
- Neste exemplo vamos analisar os dados da base Auto, disponíveis na biblioteca ISLR do R.
- A base de dados dispõe de informações técnicas de 392 modelos de automóveis das décadas de 1970 e 1980, como consumo de combustível. potência do motor, dimensões dentre outras.
- O objetivo aqui é ajustar modelos de regressão linear que expliquem o consumo de combustível com base em características do modelo.
- Neste primeiro momento, para fins ilustrativos vamos considerar apenas duas variáveis de cada vez, embora o mais usual seja considerar múltiplas variáveis conjuntamente numa análise de regressão.
summary(Auto)## mpg cylinders displacement horsepower weight
## Min. : 9.00 Min. :3.000 Min. : 68.0 Min. : 46.0 Min. :1613
## 1st Qu.:17.00 1st Qu.:4.000 1st Qu.:105.0 1st Qu.: 75.0 1st Qu.:2225
## Median :22.75 Median :4.000 Median :151.0 Median : 93.5 Median :2804
## Mean :23.45 Mean :5.472 Mean :194.4 Mean :104.5 Mean :2978
## 3rd Qu.:29.00 3rd Qu.:8.000 3rd Qu.:275.8 3rd Qu.:126.0 3rd Qu.:3615
## Max. :46.60 Max. :8.000 Max. :455.0 Max. :230.0 Max. :5140
##
## acceleration year origin name
## Min. : 8.00 Min. :70.00 Min. :1.000 amc matador : 5
## 1st Qu.:13.78 1st Qu.:73.00 1st Qu.:1.000 ford pinto : 5
## Median :15.50 Median :76.00 Median :1.000 toyota corolla : 5
## Mean :15.54 Mean :75.98 Mean :1.577 amc gremlin : 4
## 3rd Qu.:17.02 3rd Qu.:79.00 3rd Qu.:2.000 amc hornet : 4
## Max. :24.80 Max. :82.00 Max. :3.000 chevrolet chevette: 4
## (Other) :365ggplot(Auto, aes(x = year)) + geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(Auto, aes(x = year, y = mpg)) + geom_point() +
stat_smooth(method = "lm") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
plot1 <- ggplot(Auto, aes(x = horsepower, y = mpg)) + geom_point() + theme_bw(base_size = 14)
plot1
plot1 + coord_trans(x="log2", y="log2")
Fig 07
Inicialmente, vamos ajustar um modelo de regressão linear que permita explicar o consumo de combustível (mpg- variável resposta) em função do ano de lançamento do modelo (yearvariável explicativa).
A Figura 7 sugere relação linear crescente entre o consumo de combustível e o ano de lançamento do modelo.
O modelo de regressão linear para esse par de variáveis fica especificado por: \[ mpg = \beta_0 + \beta_1 \times year + \epsilon, \] onde \(\beta_0\) e \(\beta_1\) são os parâmetros do modelo (intercepto e inclinação da reta de regressão) e \(\epsilon\) representa os erros aleatórios.
O ajuste da regressão linear consiste na estimação dos parâmetros do modelo (β0 e β1), com base nos dados amostrais, que produzem a reta de regressão que melhor se ajusta aos dados.
O método usual de estimação dos parâmetros de uma regressão linear é o método de mínimos quadrados, que será estudado adiante.
Aplicando o método de mínimos quadrados, obtemos os parâmetros estimados \(\hat{\beta_0} = −70.01\) e \(\hat{\beta_1} = 1.23\), produzindo a seguinte reta de regressão ajustada: \[ \hat{mpg} = −70.01 + 1.23 \times year \]
Fig 08
Fig 09
[continua….]
Exemplo prático- Taxa de formados
- Na parte prática desta aula vamos analisar dados sobre a taxa de formados em \(n=777\) universidades norte-americanas no ano de 1995.
- O objetivo é ajustar um modelo de regressão linear que permita explicar a taxa de alunos formados em função das características das universidades e de seus alunos.
- Uma breve descrição das variáveis explicativas é apresentada na sequência.
Variaveis
Grad.Rate: Taxa de alunos formados (resposta);Apps: Número de alunos inscritos;Accept: Número de alunos aceitos;Enroll: Número de novos alunos matriculados;Top10perc: Percentual de novos estudantes entre os 10% melhores no ensino médio;Top25perc: Percentual de novos estudantes entre os 25% melhores no ensino médio;F.Undergrad: Número de alunos em período integral nos cursos de graduação;P.Undergrad: Número de alunos em período parcial nos cursos de graduação;Outstate: Número de alunos bolsistas de outros estados;Room.Board: Gastos de hospedagem e alimentação;Books: Gastos com materiais bibliográficos;Personal: Gastos com recursos humanos;PhD: Percentual de professores com doutorado;Terminal: Percentual de professores com grau terminal;S.F.Ratio: Razão alunos/professor;perc.alumni: Percentual de ex-alunos que contribuem com donativos;Expend: Gasto educacional por aluno.- Privada ou publica
CODIGO
### Carregamento e visualização inicial da base
data("College") ### Carregando a base
#help("College") ### Acessando a documentação
head(College,10) ### Visualizando as dez primeiras linhas## Private Apps Accept Enroll Top10perc Top25perc
## Abilene Christian University Yes 1660 1232 721 23 52
## Adelphi University Yes 2186 1924 512 16 29
## Adrian College Yes 1428 1097 336 22 50
## Agnes Scott College Yes 417 349 137 60 89
## Alaska Pacific University Yes 193 146 55 16 44
## Albertson College Yes 587 479 158 38 62
## Albertus Magnus College Yes 353 340 103 17 45
## Albion College Yes 1899 1720 489 37 68
## Albright College Yes 1038 839 227 30 63
## Alderson-Broaddus College Yes 582 498 172 21 44
## F.Undergrad P.Undergrad Outstate Room.Board Books
## Abilene Christian University 2885 537 7440 3300 450
## Adelphi University 2683 1227 12280 6450 750
## Adrian College 1036 99 11250 3750 400
## Agnes Scott College 510 63 12960 5450 450
## Alaska Pacific University 249 869 7560 4120 800
## Albertson College 678 41 13500 3335 500
## Albertus Magnus College 416 230 13290 5720 500
## Albion College 1594 32 13868 4826 450
## Albright College 973 306 15595 4400 300
## Alderson-Broaddus College 799 78 10468 3380 660
## Personal PhD Terminal S.F.Ratio perc.alumni Expend
## Abilene Christian University 2200 70 78 18.1 12 7041
## Adelphi University 1500 29 30 12.2 16 10527
## Adrian College 1165 53 66 12.9 30 8735
## Agnes Scott College 875 92 97 7.7 37 19016
## Alaska Pacific University 1500 76 72 11.9 2 10922
## Albertson College 675 67 73 9.4 11 9727
## Albertus Magnus College 1500 90 93 11.5 26 8861
## Albion College 850 89 100 13.7 37 11487
## Albright College 500 79 84 11.3 23 11644
## Alderson-Broaddus College 1800 40 41 11.5 15 8991
## Grad.Rate
## Abilene Christian University 60
## Adelphi University 56
## Adrian College 54
## Agnes Scott College 59
## Alaska Pacific University 15
## Albertson College 55
## Albertus Magnus College 63
## Albion College 73
## Albright College 80
## Alderson-Broaddus College 52dim(College) ### Acessando a dimensão da base## [1] 777 18summary(College) ### Resumo das variáveis## Private Apps Accept Enroll Top10perc
## No :212 Min. : 81 Min. : 72 Min. : 35 Min. : 1.00
## Yes:565 1st Qu.: 776 1st Qu.: 604 1st Qu.: 242 1st Qu.:15.00
## Median : 1558 Median : 1110 Median : 434 Median :23.00
## Mean : 3002 Mean : 2019 Mean : 780 Mean :27.56
## 3rd Qu.: 3624 3rd Qu.: 2424 3rd Qu.: 902 3rd Qu.:35.00
## Max. :48094 Max. :26330 Max. :6392 Max. :96.00
## Top25perc F.Undergrad P.Undergrad Outstate
## Min. : 9.0 Min. : 139 Min. : 1.0 Min. : 2340
## 1st Qu.: 41.0 1st Qu.: 992 1st Qu.: 95.0 1st Qu.: 7320
## Median : 54.0 Median : 1707 Median : 353.0 Median : 9990
## Mean : 55.8 Mean : 3700 Mean : 855.3 Mean :10441
## 3rd Qu.: 69.0 3rd Qu.: 4005 3rd Qu.: 967.0 3rd Qu.:12925
## Max. :100.0 Max. :31643 Max. :21836.0 Max. :21700
## Room.Board Books Personal PhD
## Min. :1780 Min. : 96.0 Min. : 250 Min. : 8.00
## 1st Qu.:3597 1st Qu.: 470.0 1st Qu.: 850 1st Qu.: 62.00
## Median :4200 Median : 500.0 Median :1200 Median : 75.00
## Mean :4358 Mean : 549.4 Mean :1341 Mean : 72.66
## 3rd Qu.:5050 3rd Qu.: 600.0 3rd Qu.:1700 3rd Qu.: 85.00
## Max. :8124 Max. :2340.0 Max. :6800 Max. :103.00
## Terminal S.F.Ratio perc.alumni Expend
## Min. : 24.0 Min. : 2.50 Min. : 0.00 Min. : 3186
## 1st Qu.: 71.0 1st Qu.:11.50 1st Qu.:13.00 1st Qu.: 6751
## Median : 82.0 Median :13.60 Median :21.00 Median : 8377
## Mean : 79.7 Mean :14.09 Mean :22.74 Mean : 9660
## 3rd Qu.: 92.0 3rd Qu.:16.50 3rd Qu.:31.00 3rd Qu.:10830
## Max. :100.0 Max. :39.80 Max. :64.00 Max. :56233
## Grad.Rate
## Min. : 10.00
## 1st Qu.: 53.00
## Median : 65.00
## Mean : 65.46
## 3rd Qu.: 78.00
## Max. :118.00### Vamos considerar Grad.Rate (taxa de formados) como a variável resposta na nossa análise. Começamos a análise com alguns gráficos.
ggplot(College, aes(x = Grad.Rate)) + geom_histogram() +
theme_bw(base_size = 14)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
### Distribuição das taxas de formados
ggplot(College, aes(x = Top10perc, y = Grad.Rate)) + geom_point() +
geom_smooth(method = "loess") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Taxas de formados versus percentual de alunos entre os 10% melhores
### no ensino médio.
ggplot(College, aes(x = Outstate, y = Grad.Rate)) + geom_point() +
geom_smooth(method = "loess") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Taxas de formados versus investimentos externos.
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
geom_smooth(method = "loess") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Taxas de formados versus porcentagens de ex-alunos contribuintes.ggpairs(College, proportions = "auto") ### Matriz de gráficos de dispersão. - DEMORA PARA RODAR
#para salvar o ultimo grafico
ggsave("plot_grande.png", device = "png", width = 100, height = 80, units = "cm")
# tamanho maximo 50 in ou 126 cm
Grafico pesado
ggcorr(College[,-1], label = TRUE, label_round = 2) ### Correlograma.
ggsave("plot_colorido.png", device = "png", width = 30, height = 20, units = "cm")
Grafico colorido
### Parte 1 - Ajuste dos modelos lineares. Comecemos com o caso de apenas uma
### variável explicativa (no caso, perc.alumni)
### Para ajustar modelos lineares no R usamos a função lm. Vamos consultar
### a documentação da função.
#help('lm')Multipla (1 var resposta) e Multivarida (mais var respostas) são coisas diferentes
### Ajuste da regressão linear simples (assumindo relação linear entre a taxa de formados e o percentual de ex-alunos contribuintes)
ajuste1 <- lm(Grad.Rate ~ perc.alumni, data = College) ###(resposta ~ var.explicativas)
ajuste1##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
##
## Coefficients:
## (Intercept) perc.alumni
## 49.9863 0.6805\[ \hat{Grad.Rate} = 49,9863 + 0,6805 \times perc.alumni \] chapeu indica estimado?
summary(ajuste1)##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -58.247 -9.513 0.043 9.362 54.404
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 49.98633 1.12345 44.49 <2e-16 ***
## perc.alumni 0.68049 0.04338 15.69 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.98 on 775 degrees of freedom
## Multiple R-squared: 0.241, Adjusted R-squared: 0.24
## F-statistic: 246.1 on 1 and 775 DF, p-value: < 2.2e-16### O percentual de ex-alunos contribuintes tem efeito positivo, e
### estatisticamente significativo na taxa de formados.### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
stat_smooth(method = "lm") +
theme_bw(base_size = 14)## `geom_smooth()` using formula = 'y ~ x'
### Vamos investigar possível efeito quadrático do percentual de contribuíntes na taxa de formados. Para isso, adicionamos ao preditor o termo quadrático da variável explicativa, da seguinte forma:
ajuste2 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
summary(ajuste2)##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2), data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -57.444 -9.146 0.042 9.448 53.446
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 45.896244 1.877789 24.442 < 2e-16 ***
## perc.alumni 1.085948 0.155614 6.978 6.42e-12 ***
## I(perc.alumni^2) -0.007652 0.002821 -2.712 0.00683 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.91 on 774 degrees of freedom
## Multiple R-squared: 0.2481, Adjusted R-squared: 0.2462
## F-statistic: 127.7 on 2 and 774 DF, p-value: < 2.2e-16### O termo quadrático é estatisticamente significativo, indicando que a
### relação entre as variáveis não é linear. Vamos dar um passo além, e
### incluir o termo de terceira ordem para o percentual de contribuintes
### (modelo cúbico).\[ \hat{Grad.Rate} = 45.896244 + 1.085948 \times perc.alumni -0.007652 \times perc.alumni^2 \]
de acordo com os valores de p é significativo, então manteria essa variavel
ajuste3 <- lm(Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3), data = College)
summary(ajuste3)##
## Call:
## lm(formula = Grad.Rate ~ perc.alumni + I(perc.alumni^2) + I(perc.alumni^3),
## data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -56.571 -9.123 0.005 9.346 52.896
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 42.9519252 2.8823421 14.902 < 2e-16 ***
## perc.alumni 1.5681680 0.3905839 4.015 6.53e-05 ***
## I(perc.alumni^2) -0.0276219 0.0151030 -1.829 0.0678 .
## I(perc.alumni^3) 0.0002297 0.0001706 1.346 0.1787
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.91 on 773 degrees of freedom
## Multiple R-squared: 0.2499, Adjusted R-squared: 0.247
## F-statistic: 85.84 on 3 and 773 DF, p-value: < 2.2e-16### O termo de ordem cúbica não tem significância estatística. Vamos seguir
### a análise com o modelo quadrático.\[ \hat{Grad.Rate} = 42.9519252 + 1.5681680 \times perc.alumni -0.0276219 \times perc.alumni^2 + 0.0002297 \times perc.alumni^3 \] de acordo com os valores de p é significativo, então manteria essa variavel
### Vamos extrair alguns elementos do modelo ajustado
ajuste2$coefficients ### Estimativas dos parâmetros## (Intercept) perc.alumni I(perc.alumni^2)
## 45.896244242 1.085947860 -0.007651754ajuste2$residuals ### Resíduos ordeinários## Abilene Christian University
## 2.17423398
## Adelphi University
## -5.31256105
## Adrian College
## -17.58810167
## Agnes Scott College
## -16.60106419
## Alaska Pacific University
## -33.03753295
## Albertson College
## -1.91580850
## Albertus Magnus College
## -5.95830307
## Albion College
## -2.60106419
## Albright College
## 13.17473271
## Alderson-Broaddus College
## -8.46381755
## Alfred University
## 0.79270745
## Allegheny College
## -1.55750846
## Allentown Coll. of St. Francis de Sales
## 8.67327410
## Alma College
## -4.81117993
## Alverno College
## -13.95830307
## American International College
## 5.23302952
## Amherst College
## 16.05885120
## Anderson University
## -0.59977055
## Andrews University
## -16.96413751
## Angelo State University
## -17.13468970
## Antioch University
## -26.53102101
## Appalachian State University
## 10.40022945
## Aquinas College
## -3.26259465
## Arizona State University Main campus
## -3.13468970
## Arkansas College (Lyon College)
## -13.55158273
## Arkansas Tech University
## -3.13468970
## Assumption College
## 16.41189833
## Auburn University-Main Campus
## 6.03586249
## Augsburg College
## -14.20729255
## Augustana College IL
## -6.09135265
## Augustana College
## -2.58810167
## Austin College
## -2.39976380
## Averett College
## -8.91580850
## Baker University
## -0.32672590
## Baldwin-Wallace College
## 20.44550005
## Barat College
## -3.53102101
## Bard College
## 7.41189833
## Barnard College
## 17.60023620
## Barry University
## 15.08419150
## Baylor University
## -4.11313051
## Beaver College
## 1.41189833
## Bellarmine College
## 11.79270745
## Belmont Abbey College
## -3.99054747
## Belmont University
## -14.76697048
## Beloit College
## 0.04169693
## Bemidji State University
## -15.31256105
## Benedictine College
## -4.96413751
## Bennington College
## -18.39976380
## Bentley College
## 17.44550005
## Berry College
## 0.85399897
## Bethany College
## -22.95360728
## Bethel College KS
## -16.81117993
## Bethel College
## -23.72042004
## Bethune Cookman College
## 2.95001707
## Birmingham-Southern College
## -1.97304415
## Blackburn College
## -30.95770455
## Bloomsburg Univ. of Pennsylvania
## 11.23302952
## Bluefield College
## 3.91477796
## Bluffton College
## -5.76697048
## Boston University
## 10.68743895
## Bowdoin College
## 14.32480917
## Bowling Green State University
## 7.40022945
## Bradford College
## -19.32672590
## Bradley University
## 19.67327410
## Brandeis University
## 16.44841727
## Brenau University
## -8.82576602
## Brewton-Parker College
## -37.99054747
## Briar Cliff College
## -10.95830307
## Bridgewater College
## 14.44841727
## Brigham Young University at Provo
## -44.09135265
## Brown University
## 20.39010666
## Bryn Mawr College
## 8.26417136
## Bucknell University
## 17.92630565
## Buena Vista College
## 22.00945253
## Butler University
## 12.04639272
## Cabrini College
## -4.07369435
## Caldwell College
## -13.26259465
## California Lutheran University
## -2.14600103
## California Polytechnic-San Luis
## 0.27957996
## California State University at Fresno
## 6.90588512
## Calvin College
## 3.44249154
## Campbell University
## -10.97304415
## Campbellsville College
## -9.72042004
## Canisius College
## -4.95830307
## Capital University
## 13.36129202
## Capitol College
## -15.55158273
## Carleton College
## 7.49319765
## Carnegie Mellon University
## 1.79270745
## Carroll College
## 10.73740535
## Carson-Newman College
## 5.68743895
## Carthage College
## -4.08364835
## Case Western Reserve University
## -3.95360728
## Castleton State College
## 24.90588512
## Catawba College
## -14.63870798
## Catholic University of America
## 12.03586249
## Cazenovia College
## 53.44550005
## Cedar Crest College
## -2.60989334
## Cedarville College
## -9.97304415
## Centenary College
## -40.55449995
## Centenary College of Louisiana
## -2.26259465
## Center for Creative Studies
## -3.11760762
## Central College
## -3.95360728
## Central Connecticut State University
## -1.11760762
## Central Missouri State University
## -0.11760762
## Central Washington University
## 5.10375576
## Central Wesleyan College
## -4.96413751
## Centre College
## -9.50680235
## Chapman University
## -5.13646827
## Chatham College
## -23.60106419
## Chestnut Hill College
## 2.04639272
## Christendom College
## -4.14600103
## Christian Brothers University
## -4.55158273
## Christopher Newport University
## -13.31256105
## Claflin College
## -51.20729255
## Claremont McKenna College
## 5.32480917
## Clark University
## 4.46897899
## Clarke College
## 4.36129202
## Clarkson University
## 4.18882007
## Clemson University
## 10.85399897
## Clinch Valley Coll. of the Univ. of Virginia
## -9.04998293
## Coe College
## -5.81117993
## Coker College
## -1.60989334
## Colby College
## 13.44249154
## Colgate University
## 13.73090339
## College Misericordia
## 29.17473271
## College of Charleston
## -11.96413751
## College of Mount St. Joseph
## 25.46897899
## College of Mount St. Vincent
## 8.46897899
## College of Notre Dame
## -10.12294333
## College of Notre Dame of Maryland
## 2.18882007
## College of Saint Benedict
## 3.04169693
## College of Saint Catherine
## 4.18882007
## College of Saint Elizabeth
## 11.17473271
## College of Saint Rose
## -6.30380939
## College of Santa Fe
## 26.87705667
## College of St. Joseph
## 12.23302952
## College of St. Scholastica
## -1.39976380
## College of the Holy Cross
## 12.52317854
## College of William and Mary
## 20.79270745
## College of Wooster
## -9.44390955
## Colorado College
## 2.62262639
## Colorado State University
## 3.00945253
## Columbia College MO
## -11.03753295
## Columbia College
## 2.02695585
## Columbia University
## 33.67327410
## Concordia College at St. Paul
## -17.96413751
## Concordia Lutheran College
## -13.04998293
## Concordia University CA
## -3.72042004
## Concordia University
## 16.27957996
## Connecticut College
## 13.90864735
## Converse College
## 2.79270745
## Cornell College
## -8.20729255
## Creighton University
## 12.18882007
## Culver-Stockton College
## -19.30380939
## Cumberland College
## 12.88239238
## D'Youville College
## -32.00836076
## Dana College
## -14.26259465
## Daniel Webster College
## 16.00945253
## Dartmouth College
## 17.26417136
## Davidson College
## 14.34126512
## Defiance College
## -7.76697048
## Delta State University
## -3.31256105
## Denison University
## 1.73090339
## DePauw University
## 9.79270745
## Dickinson College
## 10.39010666
## Dickinson State University
## -24.30380939
## Dillard University
## 3.17423398
## Doane College
## -18.00836076
## Dominican College of Blauvelt
## 2.86531030
## Dordt College
## 24.85399897
## Dowling College
## -11.12294333
## Drake University
## 9.44841727
## Drew University
## 12.69619061
## Drury College
## -7.53102101
## Duke University
## 18.13584517
## Earlham College
## -1.65873488
## East Carolina University
## -4.96413751
## East Tennessee State University
## -13.04998293
## East Texas Baptist University
## -15.12294333
## Eastern College
## -12.08364835
## Eastern Connecticut State University
## -9.59977055
## Eastern Illinois University
## 24.86531030
## Eastern Mennonite College
## 11.04639272
## Eastern Nazarene College
## 7.85399897
## Eckerd College
## -9.95830307
## Elizabethtown College
## 12.73740535
## Elmira College
## -1.32672590
## Elms College
## 20.67327410
## Elon College
## -10.97304415
## Embry Riddle Aeronautical University
## -6.11760762
## Emory & Henry College
## 0.62262639
## Emory University
## 25.69619061
## Emporia State University
## -0.11760762
## Erskine College
## -13.03306964
## Eureka College
## -19.20729255
## Evergreen State College
## 8.40022945
## Fairfield University
## 22.41189833
## Fayetteville State University
## -31.99054747
## Ferrum College
## -33.04998293
## Flagler College
## 13.95001707
## Florida Institute of Technology
## 3.87705667
## Florida International University
## 1.44550005
## Florida Southern College
## 12.00945253
## Florida State University
## -2.46381755
## Fontbonne College
## -5.55158273
## Fordham University
## 20.40022945
## Fort Lewis College
## -6.13646827
## Francis Marion University
## -11.09411488
## Franciscan University of Steubenville
## 28.90588512
## Franklin College
## -15.60106419
## Franklin Pierce College
## -10.31256105
## Freed-Hardeman University
## 3.27957996
## Fresno Pacific College
## -5.59977055
## Furman University
## 11.69619061
## Gannon University
## 2.03586249
## Gardner Webb University
## -28.82576602
## Geneva College
## 5.04169693
## George Fox College
## -14.08364835
## George Mason University
## -7.12294333
## George Washington University
## 11.53618245
## Georgetown College
## -15.30380939
## Georgetown University
## 25.36129202
## Georgia Institute of Technology
## -3.39976380
## Georgia State University
## -21.99054747
## Georgian Court College
## 6.36129202
## Gettysburg College
## 10.18882007
## Goldey Beacom College
## -14.11760762
## Gonzaga University
## -3.81117993
## Gordon College
## -6.81117993
## Goshen College
## -7.65873488
## Goucher College
## 3.02695585
## Grace College and Seminary
## -4.95830307
## Graceland College
## -20.55158273
## Grand Valley State University
## 1.95001707
## Green Mountain College
## 8.44841727
## Greensboro College
## -33.20729255
## Greenville College
## -1.31256105
## Grinnell College
## 0.77508524
## Grove City College
## 37.03586249
## Guilford College
## -6.58810167
## Gustavus Adolphus College
## -3.14072052
## Gwynedd Mercy College
## 29.91635165
## Hamilton College
## 7.49319765
## Hamline University
## -8.39976380
## Hampden - Sydney College
## -12.95770455
## Hampton University
## 14.95001707
## Hanover College
## -4.95830307
## Hardin-Simmons University
## -17.99054747
## Harding University
## -2.60106419
## Hartwick College
## 0.18882007
## Harvard University
## 18.32480917
## Harvey Mudd College
## 20.34126512
## Hastings College
## -10.14600103
## Hendrix College
## -5.95830307
## Hillsdale College
## 6.79270745
## Hiram College
## 5.02695585
## Hobart and William Smith Colleges
## 3.39893581
## Hofstra University
## 4.00945253
## Hollins College
## -8.39210089
## Hood College
## -1.97304415
## Hope College
## -5.09135265
## Houghton College
## -0.55158273
## Huntingdon College
## -11.04998293
## Huntington College
## -21.26259465
## Huron University
## -29.11760762
## Husson College
## 33.88239238
## Illinois Benedictine College
## 4.04639272
## Illinois College
## -19.58810167
## Illinois Institute of Technology
## -12.95830307
## Illinois State University
## -7.31256105
## Illinois Wesleyan University
## 9.02695585
## Immaculata College
## -4.39976380
## Incarnate Word College
## 29.67327410
## Indiana State University
## -14.09411488
## Indiana University at Bloomington
## 0.44841727
## Indiana Wesleyan University
## -26.46381755
## Iona College
## 6.40022945
## Iowa State University
## -1.08364835
## Ithaca College
## 6.73740535
## James Madison University
## 27.04639272
## Jamestown College
## -11.32672590
## Jersey City State College
## -29.99054747
## John Brown University
## 11.23302952
## John Carroll University
## 18.69619061
## Johns Hopkins University
## 13.88686949
## Johnson State College
## -24.46381755
## Judson College
## -15.58810167
## Juniata College
## 4.39893581
## Kansas State University
## -12.08364835
## Kansas Wesleyan University
## 8.40022945
## Keene State College
## 2.27957996
## Kentucky Wesleyan College
## -10.81117993
## Kenyon College
## 8.34126512
## Keuka College
## -27.44390955
## King's College
## 11.39893581
## King College
## -3.26259465
## Knox College
## 5.60023620
## La Roche College
## -7.59977055
## La Salle University
## 28.95001707
## Lafayette College
## 15.88686949
## LaGrange College
## 17.17423398
## Lake Forest College
## 13.23302952
## Lakeland College
## -11.26259465
## Lamar University
## -31.82576602
## Lambuth University
## 4.00945253
## Lander University
## -5.91580850
## Lawrence University
## -5.93472434
## Le Moyne College
## 23.69619061
## Lebanon Valley College
## 13.41189833
## Lehigh University
## 12.55609045
## Lenoir-Rhyne College
## 1.44550005
## Lesley College
## -4.96413751
## LeTourneau University
## -10.82526729
## Lewis and Clark College
## 3.67327410
## Lewis University
## 5.00945253
## Lincoln Memorial University
## -21.53102101
## Lincoln University
## -9.09411488
## Lindenwood College
## 44.95001707
## Linfield College
## 7.02695585
## Livingstone College
## 30.68743895
## Lock Haven University of Pennsylvania
## 3.40022945
## Longwood College
## -4.82526729
## Loras College
## 2.44841727
## Louisiana College
## 2.08419150
## Louisiana State University at Baton Rouge
## -19.91580850
## Louisiana Tech University
## -13.72042004
## Loyola College
## 10.36129202
## Loyola Marymount University
## 28.00945253
## Loyola University
## -6.59977055
## Loyola University Chicago
## 4.53618245
## Luther College
## 0.88686949
## Lycoming College
## -0.81117993
## Lynchburg College
## 2.44841727
## Lyndon State College
## -9.46381755
## Macalester College
## 1.39893581
## MacMurray College
## -14.39976380
## Malone College
## -6.31256105
## Manchester College
## -0.55449995
## Manhattan College
## 10.73740535
## Manhattanville College
## 2.44841727
## Mankato State University
## 11.08419150
## Marian College of Fond du Lac
## -6.32672590
## Marietta College
## 24.41189833
## Marist College
## -4.97304415
## Marquette University
## 8.73740535
## Marshall University
## -5.99054747
## Mary Baldwin College
## 8.93574712
## Mary Washington College
## 12.41189833
## Marymount College Tarrytown
## 5.41189833
## Marymount Manhattan College
## 3.44550005
## Marymount University
## -7.14600103
## Maryville College
## -27.44390955
## Maryville University
## 2.27957996
## Marywood College
## -5.58810167
## Massachusetts Institute of Technology
## 19.46897899
## Mayville State University
## -5.91580850
## McKendree College
## -13.32672590
## McMurry University
## -24.91580850
## McPherson College
## -31.26909661
## Mercer University
## 30.53618245
## Mercyhurst College
## 7.04639272
## Meredith College
## 8.60023620
## Merrimack College
## 13.91635165
## Mesa State College
## 1.17423398
## Messiah College
## 17.41189833
## Miami University at Oxford
## 20.44550005
## Michigan State University
## 15.95001707
## Michigan Technological University
## -3.26259465
## MidAmerica Nazarene College
## -15.55449995
## Millersville University of Penn.
## 6.44550005
## Milligan College
## 2.68743895
## Millikin University
## 6.73740535
## Millsaps College
## -18.11313051
## Milwaukee School of Engineering
## 0.17473271
## Mississippi College
## -1.96413751
## Mississippi State University
## -11.55449995
## Mississippi University for Women
## 8.90588512
## Missouri Southern State College
## 44.95001707
## Missouri Valley College
## -34.31256105
## Monmouth College IL
## -22.44390955
## Monmouth College
## -6.46381755
## Montana College of Mineral Sci. & Tech.
## 1.79270745
## Montana State University
## -17.09411488
## Montclair State University
## 2.95001707
## Montreat-Anderson College
## -35.11760762
## Moorhead State University
## -3.76697048
## Moravian College
## 3.69619061
## Morehouse College
## 27.00945253
## Morningside College
## -16.81117993
## Morris College
## -13.97304415
## Mount Holyoke College
## 2.62262639
## Mount Marty College
## -32.11313051
## Mount Mary College
## 9.04169693
## Mount Mercy College
## -7.58810167
## Mount Saint Clare College
## -57.44390955
## Mount Saint Mary's College
## 4.92630565
## Mount Saint Mary College
## 21.17473271
## Mount St. Mary's College
## 8.23302952
## Mount Union College
## 6.46897899
## Mount Vernon Nazarene College
## 4.87705667
## Muhlenberg College
## 6.39010666
## Murray State University
## -17.63870798
## Muskingum College
## 3.36129202
## National-Louis University
## 22.96246705
## Nazareth College of Rochester
## -6.55158273
## New Jersey Institute of Technology
## 8.23302952
## New Mexico Institute of Mining and Tech.
## -22.91580850
## New York University
## 9.68743895
## Newberry College
## 10.18882007
## Niagara University
## 0.44550005
## North Adams State College
## -5.14600103
## North Carolina A. & T. State University
## -11.04998293
## North Carolina State University at Raleigh
## -3.32672590
## North Carolina Wesleyan College
## -4.91580850
## North Central College
## 2.60023620
## North Dakota State University
## -25.55158273
## North Park College
## 6.44841727
## Northeast Missouri State University
## 17.27957996
## Northeastern University
## -16.14600103
## Northern Arizona University
## -12.12294333
## Northern Illinois University
## -0.91580850
## Northwest Missouri State University
## -12.82526729
## Northwest Nazarene College
## -6.55449995
## Northwestern College
## -5.97304415
## Northwestern University
## 23.73740535
## Norwich University
## -3.08364835
## Notre Dame College
## -35.95830307
## Oakland University
## -5.72042004
## Oberlin College
## 2.96693036
## Occidental College
## 7.41189833
## Oglethorpe University
## -2.63870798
## Ohio Northern University
## 10.79270745
## Ohio University
## 5.27957996
## Ohio Wesleyan University
## 2.18882007
## Oklahoma Baptist University
## -12.96413751
## Oklahoma Christian University
## -9.09411488
## Oklahoma State University
## -11.59977055
## Otterbein College
## 15.41189833
## Ouachita Baptist University
## 9.00945253
## Our Lady of the Lake University
## -31.26259465
## Pace University
## 6.00945253
## Pacific Lutheran University
## 16.17473271
## Pacific Union College
## 11.17423398
## Pacific University
## -24.08364835
## Pembroke State University
## -3.13468970
## Pennsylvania State Univ. Main Campus
## -0.76697048
## Pepperdine University
## 7.27957996
## Peru State College
## -23.55158273
## Pfeiffer College
## -10.72042004
## Philadelphia Coll. of Textiles and Sci.
## 7.53618245
## Phillips University
## -24.76697048
## Piedmont College
## -31.14600103
## Pikeville College
## 1.40022945
## Pitzer College
## 16.08419150
## Point Loma Nazarene College
## -9.76697048
## Point Park College
## 10.00945253
## Polytechnic University
## 2.40022945
## Prairie View A. and M. University
## -11.97454035
## Presbyterian College
## 6.99163924
## Princeton University
## 16.77508524
## Providence College
## 21.46897899
## Purdue University at West Lafayette
## 6.53618245
## Queens College
## -17.07369435
## Quincy University
## 7.18882007
## Quinnipiac College
## 12.60023620
## Radford University
## 6.95001707
## Ramapo College of New Jersey
## -7.09411488
## Randolph-Macon College
## -2.11313051
## Randolph-Macon Woman's College
## 0.44841727
## Reed College
## -7.60106419
## Regis College
## 8.39893581
## Rensselaer Polytechnic Institute
## 4.67327410
## Rhodes College
## -3.03306964
## Rider University
## 3.17473271
## Ripon College
## -16.73582864
## Rivier College
## 17.23302952
## Roanoke College
## 3.04169693
## Rockhurst College
## 13.67327410
## Rocky Mountain College
## -1.63870798
## Roger Williams University
## 6.90588512
## Rollins College
## 23.17473271
## Rosary College
## 9.41189833
## Rowan College of New Jersey
## -1.13646827
## Rutgers at New Brunswick
## 13.23302952
## Rutgers State University at Camden
## -0.82576602
## Rutgers State University at Newark
## -3.31256105
## Sacred Heart University
## 20.68743895
## Saint Ambrose University
## 8.68743895
## Saint Anselm College
## 26.04639272
## Saint Cloud State University
## 2.00945253
## Saint Francis College IN
## -2.11760762
## Saint Francis College
## 1.44841727
## Saint John's University
## -6.11313051
## Saint Joseph's College IN
## 3.23302952
## Saint Joseph's College
## 30.90588512
## Saint Joseph's University
## 24.27957996
## Saint Joseph College
## 3.18882007
## Saint Louis University
## 3.23302952
## Saint Mary's College
## 25.79270745
## Saint Mary's College of Minnesota
## 14.23302952
## Saint Mary-of-the-Woods College
## -0.60106419
## Saint Michael's College
## 10.02695585
## Saint Olaf College
## 10.79270745
## Saint Peter's College
## -13.08364835
## Saint Vincent College
## 15.79270745
## Saint Xavier University
## -5.46381755
## Salem-Teikyo University
## 42.95001707
## Salem College
## -19.65873488
## Salisbury State University
## -6.96413751
## Samford University
## -1.14600103
## San Diego State University
## -12.12294333
## Santa Clara University
## 36.23302952
## Sarah Lawrence College
## 20.03586249
## Savannah Coll. of Art and Design
## -13.95830307
## Schreiner College
## -14.82526729
## Scripps College
## -4.55750846
## Seattle Pacific University
## 1.44550005
## Seattle University
## 27.68743895
## Seton Hall University
## 3.53618245
## Seton Hill College
## -4.60106419
## Shippensburg University of Penn.
## 13.27957996
## Shorter College
## -4.96413751
## Siena College
## 21.99163924
## Siena Heights College
## -15.14600103
## Simmons College
## 5.60023620
## Simpson College
## -5.07369435
## Sioux Falls College
## -5.12294333
## Skidmore College
## 10.04639272
## Smith College
## 11.13584517
## South Dakota State University
## -3.95360728
## Southeast Missouri State University
## -9.09411488
## Southeastern Oklahoma State Univ.
## -2.04998293
## Southern California College
## -13.91580850
## Southern Illinois University at Edwardsville
## -11.09411488
## Southern Methodist University
## 9.85399897
## Southwest Baptist University
## 12.27957996
## Southwest Missouri State University
## -0.91580850
## Southwest State University
## -21.20729255
## Southwestern Adventist College
## -21.82576602
## Southwestern College
## -5.82576602
## Southwestern University
## -7.53102101
## Spalding University
## 11.90864735
## Spelman College
## 2.03586249
## Spring Arbor College
## -6.04998293
## St. Bonaventure University
## 5.18882007
## St. John's College
## -4.95830307
## St. John Fisher College
## -4.95360728
## St. Lawrence University
## 8.88686949
## St. Martin's College
## -14.09411488
## St. Mary's College of California
## 15.85399897
## St. Mary's College of Maryland
## -3.82526729
## St. Mary's University of San Antonio
## 18.87705667
## St. Norbert College
## 12.92630565
## St. Paul's College
## -9.09411488
## St. Thomas Aquinas College
## 8.27957996
## Stephens College
## -3.14600103
## Stetson University
## 5.44841727
## Stevens Institute of Technology
## 5.60023620
## Stockton College of New Jersey
## 10.87705667
## Stonehill College
## 25.41189833
## SUNY at Albany
## 12.68743895
## SUNY at Binghamton
## 19.53618245
## SUNY at Buffalo
## -4.46381755
## SUNY at Stony Brook
## 3.87705667
## SUNY College at Brockport
## -10.59977055
## SUNY College at Oswego
## -2.32672590
## SUNY College at Buffalo
## -15.82576602
## SUNY College at Cortland
## -9.14600103
## SUNY College at Fredonia
## 10.00945253
## SUNY College at Geneseo
## 7.73740535
## SUNY College at New Paltz
## -1.09411488
## SUNY College at Plattsburgh
## 3.68743895
## SUNY College at Potsdam
## -3.14600103
## SUNY College at Purchase
## -8.09411488
## Susquehanna University
## 14.39893581
## Sweet Briar College
## -19.39210089
## Syracuse University
## 8.27957996
## Tabor College
## -7.46381755
## Talladega College
## -7.12294333
## Taylor University
## 25.18882007
## Tennessee Wesleyan College
## -25.31256105
## Texas A&M Univ. at College Station
## -1.95360728
## Texas A&M University at Galveston
## -18.31256105
## Texas Christian University
## -2.82526729
## Texas Lutheran College
## -17.55158273
## Texas Southern University
## -55.32672590
## Texas Wesleyan University
## -12.99054747
## The Citadel
## 21.85399897
## Thiel College
## -1.31256105
## Tiffin University
## -38.09135265
## Transylvania University
## -7.55750846
## Trenton State College
## 28.86353173
## Tri-State University
## -2.55158273
## Trinity College CT
## 10.60789911
## Trinity College DC
## 20.39893581
## Trinity College VT
## 9.04169693
## Trinity University
## 28.44550005
## Tulane University
## 8.67327410
## Tusculum College
## -18.30380939
## Tuskegee University
## 11.87705667
## Union College KY
## 8.95001707
## Union College NY
## 7.26417136
## Univ. of Wisconsin at OshKosh
## -3.59977055
## University of Alabama at Birmingham
## -28.31256105
## University of Arkansas at Fayetteville
## -16.99054747
## University of California at Berkeley
## 22.00945253
## University of California at Irvine
## 9.08419150
## University of Central Florida
## -9.04998293
## University of Charleston
## 1.00945253
## University of Chicago
## 14.92630565
## University of Cincinnati
## 1.86353173
## University of Connecticut at Storrs
## 9.68743895
## University of Dallas
## -5.95830307
## University of Dayton
## 24.73740535
## University of Delaware
## 14.53618245
## University of Denver
## 1.67327410
## University of Detroit Mercy
## -8.59977055
## University of Dubuque
## -17.96413751
## University of Evansville
## 8.04169693
## University of Florida
## 1.44550005
## University of Georgia
## -3.08364835
## University of Hartford
## 10.95001707
## University of Hawaii at Manoa
## 1.86353173
## University of Illinois - Urbana
## 22.27957996
## University of Illinois at Chicago
## -18.13646827
## University of Indianapolis
## 2.17473271
## University of Kansas
## -5.14600103
## University of La Verne
## -19.82526729
## University of Louisville
## -36.55158273
## University of Maine at Farmington
## -9.95830307
## University of Maine at Machias
## 13.88239238
## University of Maine at Presque Isle
## -21.91580850
## University of Maryland at Baltimore County
## 2.86353173
## University of Maryland at College Park
## 5.17423398
## University of Massachusetts at Amherst
## 7.53618245
## University of Massachusetts at Dartmouth
## -8.55449995
## University of Miami
## -3.14600103
## University of Michigan at Ann Arbor
## 18.04169693
## University of Minnesota at Duluth
## -3.91580850
## University of Minnesota at Morris
## -10.31256105
## University of Minnesota Twin Cities
## -30.60106419
## University of Mississippi
## -6.59977055
## University of Missouri at Columbia
## -2.46381755
## University of Missouri at Rolla
## -17.82526729
## University of Missouri at Saint Louis
## -12.46381755
## University of Mobile
## 1.88239238
## University of Montevallo
## -3.09411488
## University of Nebraska at Lincoln
## -27.39210089
## University of New England
## 5.27957996
## University of New Hampshire
## 13.68743895
## University of North Carolina at Asheville
## -19.91580850
## University of North Carolina at Chapel Hill
## 16.17473271
## University of North Carolina at Charlotte
## 8.87705667
## University of North Carolina at Greensboro
## -9.14600103
## University of North Carolina at Wilmington
## -5.46381755
## University of North Dakota
## -12.31256105
## University of North Florida
## -12.59977055
## University of North Texas
## -17.13646827
## University of Northern Colorado
## -14.09411488
## University of Northern Iowa
## 8.04169693
## University of Notre Dame
## 17.34126512
## University of Oklahoma
## -12.91580850
## University of Oregon
## -4.72042004
## University of Pennsylvania
## 16.88686949
## University of Pittsburgh-Main Campus
## 10.00945253
## University of Portland
## 9.85399897
## University of Puget Sound
## 0.85399897
## University of Rhode Island
## 9.87705667
## University of Richmond
## 27.18882007
## University of Rochester
## 13.17473271
## University of San Diego
## 7.27957996
## University of San Francisco
## 7.90588512
## University of Sci. and Arts of Oklahoma
## -6.08522204
## University of Scranton
## 14.44249154
## University of South Carolina at Aiken
## -1.08522204
## University of South Carolina at Columbia
## 0.03586249
## University of South Florida
## -6.12294333
## University of Southern California
## 12.00945253
## University of Southern Colorado
## -9.89624424
## University of Southern Indiana
## -27.32672590
## University of Southern Mississippi
## -21.82526729
## University of St. Thomas MN
## 30.27957996
## University of St. Thomas TX
## -0.14600103
## University of Tennessee at Knoxville
## -13.08364835
## University of Texas at Arlington
## -21.11760762
## University of Texas at Austin
## 8.08419150
## University of Texas at San Antonio
## 0.91477796
## University of the Arts
## 1.95001707
## University of the Pacific
## 5.40022945
## University of the South
## 0.32480917
## University of Tulsa
## -8.99054747
## University of Utah
## -18.04998293
## University of Vermont
## 23.00945253
## University of Virginia
## 28.91635165
## University of Washington
## 9.00945253
## University of West Florida
## -4.82576602
## University of Wisconsin-Stout
## 2.85399897
## University of Wisconsin-Superior
## -24.46381755
## University of Wisconsin-Whitewater
## 5.68743895
## University of Wisconsin at Green Bay
## -0.97454035
## University of Wisconsin at Madison
## 7.44550005
## University of Wisconsin at Milwaukee
## -16.09411488
## University of Wyoming
## -13.72042004
## Upper Iowa University
## 14.23302952
## Ursinus College
## 1.90864735
## Ursuline College
## 7.53618245
## Valley City State University
## -28.26259465
## Valparaiso University
## 28.17473271
## Vanderbilt University
## 14.04169693
## Vassar College
## 13.39010666
## Villanova University
## 28.44841727
## Virginia Commonwealth University
## -11.91580850
## Virginia State University
## -25.91580850
## Virginia Tech
## 8.44550005
## Virginia Union University
## -24.09411488
## Virginia Wesleyan College
## 8.40022945
## Viterbo College
## 0.79270745
## Voorhees College
## 8.91477796
## Wabash College
## -10.47682146
## Wagner College
## 8.17473271
## Wake Forest University
## 13.39893581
## Walsh University
## -5.39976380
## Warren Wilson College
## -1.55449995
## Wartburg College
## -8.60106419
## Washington and Jefferson College
## 8.90864735
## Washington and Lee University
## 10.73090339
## Washington College
## -10.60106419
## Washington State University
## -15.58810167
## Washington University
## 17.79270745
## Wayne State College
## -18.95360728
## Waynesburg College
## -5.95830307
## Webber College
## 39.88239238
## Webster University
## -11.59977055
## Wellesley College
## 9.62262639
## Wells College
## -9.00836076
## Wentworth Institute of Technology
## 9.90588512
## Wesley College
## 23.53618245
## Wesleyan University
## 15.39010666
## West Chester University of Penn.
## -9.31256105
## West Liberty State College
## 4.00945253
## West Virginia Wesleyan College
## -11.00836076
## Western Carolina University
## -0.04998293
## Western Maryland College
## -16.60989334
## Western Michigan University
## -1.91580850
## Western New England College
## -1.46381755
## Western State College of Colorado
## 1.88239238
## Western Washington University
## 5.00945253
## Westfield State College
## 0.44550005
## Westminster College MO
## -2.55449995
## Westminster College
## 2.44249154
## Westminster College of Salt Lake City
## -23.97304415
## Westmont College
## 24.85399897
## Wheaton College IL
## 7.90864735
## Westminster College PA
## -6.55750846
## Wheeling Jesuit College
## 2.36129202
## Whitman College
## -9.37737361
## Whittier College
## -18.95360728
## Whitworth College
## 15.44550005
## Widener University
## -0.76697048
## Wilkes University
## -0.55158273
## Willamette University
## -7.60106419
## William Jewell College
## -4.76697048
## William Woods University
## -9.31256105
## Williams College
## 14.94467606
## Wilson College
## -11.44390955
## Wingate College
## 36.90588512
## Winona State University
## -4.96413751
## Winthrop University
## -9.95830307
## Wisconsin Lutheran College
## -18.95830307
## Wittenberg University
## 7.04639272
## Wofford College
## -3.00836076
## Worcester Polytechnic Institute
## 8.02695585
## Worcester State College
## -19.59977055
## Xavier University
## 10.79270745
## Xavier University of Louisiana
## -15.55449995
## Yale University
## 18.26417136
## York College of Pennsylvania
## 28.69619061ajuste2$fitted.values ### Valores ajustados pelo modelo## Abilene Christian University
## 57.82577
## Adelphi University
## 61.31256
## Adrian College
## 71.58810
## Agnes Scott College
## 75.60106
## Alaska Pacific University
## 48.03753
## Albertson College
## 56.91581
## Albertus Magnus College
## 68.95830
## Albion College
## 75.60106
## Albright College
## 66.82527
## Alderson-Broaddus College
## 60.46382
## Alfred University
## 72.20729
## Allegheny College
## 77.55751
## Allentown Coll. of St. Francis de Sales
## 65.32673
## Alma College
## 72.81118
## Alverno College
## 68.95830
## American International College
## 63.76697
## Amherst College
## 83.94115
## Anderson University
## 59.59977
## Andrews University
## 62.96414
## Angelo State University
## 51.13469
## Antioch University
## 74.53102
## Appalachian State University
## 59.59977
## Aquinas College
## 68.26259
## Arizona State University Main campus
## 51.13469
## Arkansas College (Lyon College)
## 67.55158
## Arkansas Tech University
## 51.13469
## Assumption College
## 71.58810
## Auburn University-Main Campus
## 62.96414
## Augsburg College
## 72.20729
## Augustana College IL
## 77.09135
## Augustana College
## 71.58810
## Austin College
## 73.39976
## Averett College
## 56.91581
## Baker University
## 65.32673
## Baldwin-Wallace College
## 64.55450
## Barat College
## 74.53102
## Bard College
## 71.58810
## Barnard College
## 73.39976
## Barry University
## 56.91581
## Baylor University
## 76.11313
## Beaver College
## 71.58810
## Bellarmine College
## 72.20729
## Belmont Abbey College
## 55.99055
## Belmont University
## 63.76697
## Beloit College
## 68.95830
## Bemidji State University
## 61.31256
## Benedictine College
## 62.96414
## Bennington College
## 73.39976
## Bentley College
## 64.55450
## Berry College
## 62.14600
## Bethany College
## 70.95361
## Bethel College KS
## 72.81118
## Bethel College
## 58.72042
## Bethune Cookman College
## 55.04998
## Birmingham-Southern College
## 73.97304
## Blackburn College
## 81.95770
## Bloomsburg Univ. of Pennsylvania
## 63.76697
## Bluefield College
## 49.08522
## Bluffton College
## 63.76697
## Boston University
## 61.31256
## Bowdoin College
## 81.67519
## Bowling Green State University
## 59.59977
## Bradford College
## 65.32673
## Bradley University
## 65.32673
## Brandeis University
## 67.55158
## Brenau University
## 57.82577
## Brewton-Parker College
## 55.99055
## Briar Cliff College
## 68.95830
## Bridgewater College
## 67.55158
## Brigham Young University at Provo
## 77.09135
## Brown University
## 76.60989
## Bryn Mawr College
## 80.73583
## Bucknell University
## 75.07369
## Buena Vista College
## 55.99055
## Butler University
## 70.95361
## Cabrini College
## 75.07369
## Caldwell College
## 68.26259
## California Lutheran University
## 62.14600
## California Polytechnic-San Luis
## 58.72042
## California State University at Fresno
## 54.09411
## Calvin College
## 77.55751
## Campbell University
## 73.97304
## Campbellsville College
## 58.72042
## Canisius College
## 68.95830
## Capital University
## 69.63871
## Capitol College
## 67.55158
## Carleton College
## 83.50680
## Carnegie Mellon University
## 72.20729
## Carroll College
## 68.26259
## Carson-Newman College
## 61.31256
## Carthage College
## 66.08365
## Case Western Reserve University
## 70.95361
## Castleton State College
## 54.09411
## Catawba College
## 69.63871
## Catholic University of America
## 62.96414
## Cazenovia College
## 64.55450
## Cedar Crest College
## 76.60989
## Cedarville College
## 73.97304
## Centenary College
## 64.55450
## Centenary College of Louisiana
## 68.26259
## Center for Creative Studies
## 50.11761
## Central College
## 70.95361
## Central Connecticut State University
## 50.11761
## Central Missouri State University
## 50.11761
## Central Washington University
## 45.89624
## Central Wesleyan College
## 62.96414
## Centre College
## 83.50680
## Chapman University
## 52.13647
## Chatham College
## 75.60106
## Chestnut Hill College
## 70.95361
## Christendom College
## 62.14600
## Christian Brothers University
## 67.55158
## Christopher Newport University
## 61.31256
## Claflin College
## 72.20729
## Claremont McKenna College
## 81.67519
## Clark University
## 74.53102
## Clarke College
## 69.63871
## Clarkson University
## 72.81118
## Clemson University
## 62.14600
## Clinch Valley Coll. of the Univ. of Virginia
## 55.04998
## Coe College
## 72.81118
## Coker College
## 76.60989
## Colby College
## 77.55751
## Colgate University
## 79.26910
## College Misericordia
## 66.82527
## College of Charleston
## 62.96414
## College of Mount St. Joseph
## 74.53102
## College of Mount St. Vincent
## 74.53102
## College of Notre Dame
## 53.12294
## College of Notre Dame of Maryland
## 72.81118
## College of Saint Benedict
## 68.95830
## College of Saint Catherine
## 72.81118
## College of Saint Elizabeth
## 66.82527
## College of Saint Rose
## 70.30381
## College of Santa Fe
## 53.12294
## College of St. Joseph
## 63.76697
## College of St. Scholastica
## 73.39976
## College of the Holy Cross
## 82.47682
## College of William and Mary
## 72.20729
## College of Wooster
## 78.44391
## Colorado College
## 81.37737
## Colorado State University
## 55.99055
## Columbia College MO
## 48.03753
## Columbia College
## 73.97304
## Columbia University
## 65.32673
## Concordia College at St. Paul
## 62.96414
## Concordia Lutheran College
## 55.04998
## Concordia University CA
## 58.72042
## Concordia University
## 58.72042
## Connecticut College
## 77.09135
## Converse College
## 72.20729
## Cornell College
## 72.20729
## Creighton University
## 72.81118
## Culver-Stockton College
## 70.30381
## Cumberland College
## 50.11761
## D'Youville College
## 78.00836
## Dana College
## 68.26259
## Daniel Webster College
## 55.99055
## Dartmouth College
## 80.73583
## Davidson College
## 79.65873
## Defiance College
## 63.76697
## Delta State University
## 61.31256
## Denison University
## 79.26910
## DePauw University
## 72.20729
## Dickinson College
## 76.60989
## Dickinson State University
## 70.30381
## Dillard University
## 57.82577
## Doane College
## 78.00836
## Dominican College of Blauvelt
## 51.13469
## Dordt College
## 62.14600
## Dowling College
## 53.12294
## Drake University
## 67.55158
## Drew University
## 70.30381
## Drury College
## 74.53102
## Duke University
## 78.86415
## Earlham College
## 79.65873
## East Carolina University
## 62.96414
## East Tennessee State University
## 55.04998
## East Texas Baptist University
## 53.12294
## Eastern College
## 66.08365
## Eastern Connecticut State University
## 59.59977
## Eastern Illinois University
## 51.13469
## Eastern Mennonite College
## 70.95361
## Eastern Nazarene College
## 62.14600
## Eckerd College
## 68.95830
## Elizabethtown College
## 68.26259
## Elmira College
## 65.32673
## Elms College
## 65.32673
## Elon College
## 73.97304
## Embry Riddle Aeronautical University
## 50.11761
## Emory & Henry College
## 81.37737
## Emory University
## 70.30381
## Emporia State University
## 50.11761
## Erskine College
## 80.03307
## Eureka College
## 72.20729
## Evergreen State College
## 59.59977
## Fairfield University
## 71.58810
## Fayetteville State University
## 55.99055
## Ferrum College
## 55.04998
## Flagler College
## 55.04998
## Florida Institute of Technology
## 53.12294
## Florida International University
## 64.55450
## Florida Southern College
## 55.99055
## Florida State University
## 60.46382
## Fontbonne College
## 67.55158
## Fordham University
## 59.59977
## Fort Lewis College
## 52.13647
## Francis Marion University
## 54.09411
## Franciscan University of Steubenville
## 54.09411
## Franklin College
## 75.60106
## Franklin Pierce College
## 61.31256
## Freed-Hardeman University
## 58.72042
## Fresno Pacific College
## 59.59977
## Furman University
## 70.30381
## Gannon University
## 62.96414
## Gardner Webb University
## 57.82577
## Geneva College
## 68.95830
## George Fox College
## 66.08365
## George Mason University
## 53.12294
## George Washington University
## 60.46382
## Georgetown College
## 70.30381
## Georgetown University
## 69.63871
## Georgia Institute of Technology
## 73.39976
## Georgia State University
## 55.99055
## Georgian Court College
## 69.63871
## Gettysburg College
## 72.81118
## Goldey Beacom College
## 50.11761
## Gonzaga University
## 72.81118
## Gordon College
## 72.81118
## Goshen College
## 79.65873
## Goucher College
## 73.97304
## Grace College and Seminary
## 68.95830
## Graceland College
## 67.55158
## Grand Valley State University
## 55.04998
## Green Mountain College
## 67.55158
## Greensboro College
## 72.20729
## Greenville College
## 61.31256
## Grinnell College
## 82.22491
## Grove City College
## 62.96414
## Guilford College
## 71.58810
## Gustavus Adolphus College
## 83.14072
## Gwynedd Mercy College
## 66.08365
## Hamilton College
## 83.50680
## Hamline University
## 73.39976
## Hampden - Sydney College
## 81.95770
## Hampton University
## 55.04998
## Hanover College
## 68.95830
## Hardin-Simmons University
## 55.99055
## Harding University
## 75.60106
## Hartwick College
## 72.81118
## Harvard University
## 81.67519
## Harvey Mudd College
## 79.65873
## Hastings College
## 62.14600
## Hendrix College
## 68.95830
## Hillsdale College
## 72.20729
## Hiram College
## 73.97304
## Hobart and William Smith Colleges
## 75.60106
## Hofstra University
## 55.99055
## Hollins College
## 80.39210
## Hood College
## 73.97304
## Hope College
## 77.09135
## Houghton College
## 67.55158
## Huntingdon College
## 55.04998
## Huntington College
## 68.26259
## Huron University
## 50.11761
## Husson College
## 50.11761
## Illinois Benedictine College
## 70.95361
## Illinois College
## 71.58810
## Illinois Institute of Technology
## 68.95830
## Illinois State University
## 61.31256
## Illinois Wesleyan University
## 73.97304
## Immaculata College
## 73.39976
## Incarnate Word College
## 65.32673
## Indiana State University
## 54.09411
## Indiana University at Bloomington
## 67.55158
## Indiana Wesleyan University
## 60.46382
## Iona College
## 59.59977
## Iowa State University
## 66.08365
## Ithaca College
## 68.26259
## James Madison University
## 70.95361
## Jamestown College
## 65.32673
## Jersey City State College
## 55.99055
## John Brown University
## 63.76697
## John Carroll University
## 70.30381
## Johns Hopkins University
## 76.11313
## Johnson State College
## 60.46382
## Judson College
## 71.58810
## Juniata College
## 75.60106
## Kansas State University
## 66.08365
## Kansas Wesleyan University
## 59.59977
## Keene State College
## 58.72042
## Kentucky Wesleyan College
## 72.81118
## Kenyon College
## 79.65873
## Keuka College
## 78.44391
## King's College
## 75.60106
## King College
## 68.26259
## Knox College
## 73.39976
## La Roche College
## 59.59977
## La Salle University
## 55.04998
## Lafayette College
## 76.11313
## LaGrange College
## 57.82577
## Lake Forest College
## 63.76697
## Lakeland College
## 68.26259
## Lamar University
## 57.82577
## Lambuth University
## 55.99055
## Lander University
## 56.91581
## Lawrence University
## 82.93472
## Le Moyne College
## 70.30381
## Lebanon Valley College
## 71.58810
## Lehigh University
## 78.44391
## Lenoir-Rhyne College
## 64.55450
## Lesley College
## 62.96414
## LeTourneau University
## 66.82527
## Lewis and Clark College
## 65.32673
## Lewis University
## 55.99055
## Lincoln Memorial University
## 74.53102
## Lincoln University
## 54.09411
## Lindenwood College
## 55.04998
## Linfield College
## 73.97304
## Livingstone College
## 61.31256
## Lock Haven University of Pennsylvania
## 59.59977
## Longwood College
## 66.82527
## Loras College
## 67.55158
## Louisiana College
## 56.91581
## Louisiana State University at Baton Rouge
## 56.91581
## Louisiana Tech University
## 58.72042
## Loyola College
## 69.63871
## Loyola Marymount University
## 55.99055
## Loyola University
## 59.59977
## Loyola University Chicago
## 60.46382
## Luther College
## 76.11313
## Lycoming College
## 72.81118
## Lynchburg College
## 67.55158
## Lyndon State College
## 60.46382
## Macalester College
## 75.60106
## MacMurray College
## 73.39976
## Malone College
## 61.31256
## Manchester College
## 64.55450
## Manhattan College
## 68.26259
## Manhattanville College
## 67.55158
## Mankato State University
## 56.91581
## Marian College of Fond du Lac
## 65.32673
## Marietta College
## 71.58810
## Marist College
## 73.97304
## Marquette University
## 68.26259
## Marshall University
## 55.99055
## Mary Baldwin College
## 81.06425
## Mary Washington College
## 71.58810
## Marymount College Tarrytown
## 71.58810
## Marymount Manhattan College
## 64.55450
## Marymount University
## 62.14600
## Maryville College
## 78.44391
## Maryville University
## 58.72042
## Marywood College
## 71.58810
## Massachusetts Institute of Technology
## 74.53102
## Mayville State University
## 56.91581
## McKendree College
## 65.32673
## McMurry University
## 56.91581
## McPherson College
## 79.26910
## Mercer University
## 60.46382
## Mercyhurst College
## 70.95361
## Meredith College
## 73.39976
## Merrimack College
## 66.08365
## Mesa State College
## 57.82577
## Messiah College
## 71.58810
## Miami University at Oxford
## 64.55450
## Michigan State University
## 55.04998
## Michigan Technological University
## 68.26259
## MidAmerica Nazarene College
## 64.55450
## Millersville University of Penn.
## 64.55450
## Milligan College
## 61.31256
## Millikin University
## 68.26259
## Millsaps College
## 76.11313
## Milwaukee School of Engineering
## 66.82527
## Mississippi College
## 62.96414
## Mississippi State University
## 64.55450
## Mississippi University for Women
## 54.09411
## Missouri Southern State College
## 55.04998
## Missouri Valley College
## 61.31256
## Monmouth College IL
## 78.44391
## Monmouth College
## 60.46382
## Montana College of Mineral Sci. & Tech.
## 72.20729
## Montana State University
## 54.09411
## Montclair State University
## 55.04998
## Montreat-Anderson College
## 50.11761
## Moorhead State University
## 63.76697
## Moravian College
## 70.30381
## Morehouse College
## 55.99055
## Morningside College
## 72.81118
## Morris College
## 73.97304
## Mount Holyoke College
## 81.37737
## Mount Marty College
## 76.11313
## Mount Mary College
## 68.95830
## Mount Mercy College
## 71.58810
## Mount Saint Clare College
## 78.44391
## Mount Saint Mary's College
## 75.07369
## Mount Saint Mary College
## 66.82527
## Mount St. Mary's College
## 63.76697
## Mount Union College
## 74.53102
## Mount Vernon Nazarene College
## 53.12294
## Muhlenberg College
## 76.60989
## Murray State University
## 69.63871
## Muskingum College
## 69.63871
## National-Louis University
## 48.03753
## Nazareth College of Rochester
## 67.55158
## New Jersey Institute of Technology
## 63.76697
## New Mexico Institute of Mining and Tech.
## 56.91581
## New York University
## 61.31256
## Newberry College
## 72.81118
## Niagara University
## 64.55450
## North Adams State College
## 62.14600
## North Carolina A. & T. State University
## 55.04998
## North Carolina State University at Raleigh
## 65.32673
## North Carolina Wesleyan College
## 56.91581
## North Central College
## 73.39976
## North Dakota State University
## 67.55158
## North Park College
## 67.55158
## Northeast Missouri State University
## 58.72042
## Northeastern University
## 62.14600
## Northern Arizona University
## 53.12294
## Northern Illinois University
## 56.91581
## Northwest Missouri State University
## 66.82527
## Northwest Nazarene College
## 64.55450
## Northwestern College
## 73.97304
## Northwestern University
## 68.26259
## Norwich University
## 66.08365
## Notre Dame College
## 68.95830
## Oakland University
## 58.72042
## Oberlin College
## 80.03307
## Occidental College
## 71.58810
## Oglethorpe University
## 69.63871
## Ohio Northern University
## 72.20729
## Ohio University
## 58.72042
## Ohio Wesleyan University
## 72.81118
## Oklahoma Baptist University
## 62.96414
## Oklahoma Christian University
## 54.09411
## Oklahoma State University
## 59.59977
## Otterbein College
## 71.58810
## Ouachita Baptist University
## 55.99055
## Our Lady of the Lake University
## 68.26259
## Pace University
## 55.99055
## Pacific Lutheran University
## 66.82527
## Pacific Union College
## 57.82577
## Pacific University
## 66.08365
## Pembroke State University
## 51.13469
## Pennsylvania State Univ. Main Campus
## 63.76697
## Pepperdine University
## 58.72042
## Peru State College
## 67.55158
## Pfeiffer College
## 58.72042
## Philadelphia Coll. of Textiles and Sci.
## 60.46382
## Phillips University
## 63.76697
## Piedmont College
## 62.14600
## Pikeville College
## 59.59977
## Pitzer College
## 56.91581
## Point Loma Nazarene College
## 63.76697
## Point Park College
## 55.99055
## Polytechnic University
## 59.59977
## Prairie View A. and M. University
## 46.97454
## Presbyterian College
## 78.00836
## Princeton University
## 82.22491
## Providence College
## 74.53102
## Purdue University at West Lafayette
## 60.46382
## Queens College
## 75.07369
## Quincy University
## 72.81118
## Quinnipiac College
## 73.39976
## Radford University
## 55.04998
## Ramapo College of New Jersey
## 54.09411
## Randolph-Macon College
## 76.11313
## Randolph-Macon Woman's College
## 67.55158
## Reed College
## 75.60106
## Regis College
## 75.60106
## Rensselaer Polytechnic Institute
## 65.32673
## Rhodes College
## 80.03307
## Rider University
## 66.82527
## Ripon College
## 80.73583
## Rivier College
## 63.76697
## Roanoke College
## 68.95830
## Rockhurst College
## 65.32673
## Rocky Mountain College
## 69.63871
## Roger Williams University
## 54.09411
## Rollins College
## 66.82527
## Rosary College
## 71.58810
## Rowan College of New Jersey
## 52.13647
## Rutgers at New Brunswick
## 63.76697
## Rutgers State University at Camden
## 57.82577
## Rutgers State University at Newark
## 61.31256
## Sacred Heart University
## 61.31256
## Saint Ambrose University
## 61.31256
## Saint Anselm College
## 70.95361
## Saint Cloud State University
## 55.99055
## Saint Francis College IN
## 50.11761
## Saint Francis College
## 67.55158
## Saint John's University
## 76.11313
## Saint Joseph's College IN
## 63.76697
## Saint Joseph's College
## 54.09411
## Saint Joseph's University
## 58.72042
## Saint Joseph College
## 72.81118
## Saint Louis University
## 63.76697
## Saint Mary's College
## 72.20729
## Saint Mary's College of Minnesota
## 63.76697
## Saint Mary-of-the-Woods College
## 75.60106
## Saint Michael's College
## 73.97304
## Saint Olaf College
## 72.20729
## Saint Peter's College
## 66.08365
## Saint Vincent College
## 72.20729
## Saint Xavier University
## 60.46382
## Salem-Teikyo University
## 55.04998
## Salem College
## 79.65873
## Salisbury State University
## 62.96414
## Samford University
## 62.14600
## San Diego State University
## 53.12294
## Santa Clara University
## 63.76697
## Sarah Lawrence College
## 62.96414
## Savannah Coll. of Art and Design
## 68.95830
## Schreiner College
## 66.82527
## Scripps College
## 77.55751
## Seattle Pacific University
## 64.55450
## Seattle University
## 61.31256
## Seton Hall University
## 60.46382
## Seton Hill College
## 75.60106
## Shippensburg University of Penn.
## 58.72042
## Shorter College
## 62.96414
## Siena College
## 78.00836
## Siena Heights College
## 62.14600
## Simmons College
## 73.39976
## Simpson College
## 75.07369
## Sioux Falls College
## 53.12294
## Skidmore College
## 70.95361
## Smith College
## 78.86415
## South Dakota State University
## 70.95361
## Southeast Missouri State University
## 54.09411
## Southeastern Oklahoma State Univ.
## 55.04998
## Southern California College
## 56.91581
## Southern Illinois University at Edwardsville
## 54.09411
## Southern Methodist University
## 62.14600
## Southwest Baptist University
## 58.72042
## Southwest Missouri State University
## 56.91581
## Southwest State University
## 72.20729
## Southwestern Adventist College
## 57.82577
## Southwestern College
## 57.82577
## Southwestern University
## 74.53102
## Spalding University
## 77.09135
## Spelman College
## 62.96414
## Spring Arbor College
## 55.04998
## St. Bonaventure University
## 72.81118
## St. John's College
## 68.95830
## St. John Fisher College
## 70.95361
## St. Lawrence University
## 76.11313
## St. Martin's College
## 54.09411
## St. Mary's College of California
## 62.14600
## St. Mary's College of Maryland
## 66.82527
## St. Mary's University of San Antonio
## 53.12294
## St. Norbert College
## 75.07369
## St. Paul's College
## 54.09411
## St. Thomas Aquinas College
## 58.72042
## Stephens College
## 62.14600
## Stetson University
## 67.55158
## Stevens Institute of Technology
## 73.39976
## Stockton College of New Jersey
## 53.12294
## Stonehill College
## 71.58810
## SUNY at Albany
## 61.31256
## SUNY at Binghamton
## 60.46382
## SUNY at Buffalo
## 60.46382
## SUNY at Stony Brook
## 53.12294
## SUNY College at Brockport
## 59.59977
## SUNY College at Oswego
## 65.32673
## SUNY College at Buffalo
## 57.82577
## SUNY College at Cortland
## 62.14600
## SUNY College at Fredonia
## 55.99055
## SUNY College at Geneseo
## 68.26259
## SUNY College at New Paltz
## 54.09411
## SUNY College at Plattsburgh
## 61.31256
## SUNY College at Potsdam
## 62.14600
## SUNY College at Purchase
## 54.09411
## Susquehanna University
## 75.60106
## Sweet Briar College
## 80.39210
## Syracuse University
## 58.72042
## Tabor College
## 60.46382
## Talladega College
## 53.12294
## Taylor University
## 72.81118
## Tennessee Wesleyan College
## 61.31256
## Texas A&M Univ. at College Station
## 70.95361
## Texas A&M University at Galveston
## 61.31256
## Texas Christian University
## 66.82527
## Texas Lutheran College
## 67.55158
## Texas Southern University
## 65.32673
## Texas Wesleyan University
## 55.99055
## The Citadel
## 62.14600
## Thiel College
## 61.31256
## Tiffin University
## 77.09135
## Transylvania University
## 77.55751
## Trenton State College
## 52.13647
## Tri-State University
## 67.55158
## Trinity College CT
## 80.39210
## Trinity College DC
## 75.60106
## Trinity College VT
## 68.95830
## Trinity University
## 64.55450
## Tulane University
## 65.32673
## Tusculum College
## 70.30381
## Tuskegee University
## 53.12294
## Union College KY
## 55.04998
## Union College NY
## 80.73583
## Univ. of Wisconsin at OshKosh
## 59.59977
## University of Alabama at Birmingham
## 61.31256
## University of Arkansas at Fayetteville
## 55.99055
## University of California at Berkeley
## 55.99055
## University of California at Irvine
## 56.91581
## University of Central Florida
## 55.04998
## University of Charleston
## 55.99055
## University of Chicago
## 75.07369
## University of Cincinnati
## 52.13647
## University of Connecticut at Storrs
## 61.31256
## University of Dallas
## 68.95830
## University of Dayton
## 68.26259
## University of Delaware
## 60.46382
## University of Denver
## 65.32673
## University of Detroit Mercy
## 59.59977
## University of Dubuque
## 62.96414
## University of Evansville
## 68.95830
## University of Florida
## 64.55450
## University of Georgia
## 66.08365
## University of Hartford
## 55.04998
## University of Hawaii at Manoa
## 52.13647
## University of Illinois - Urbana
## 58.72042
## University of Illinois at Chicago
## 52.13647
## University of Indianapolis
## 66.82527
## University of Kansas
## 62.14600
## University of La Verne
## 66.82527
## University of Louisville
## 67.55158
## University of Maine at Farmington
## 68.95830
## University of Maine at Machias
## 50.11761
## University of Maine at Presque Isle
## 56.91581
## University of Maryland at Baltimore County
## 52.13647
## University of Maryland at College Park
## 57.82577
## University of Massachusetts at Amherst
## 60.46382
## University of Massachusetts at Dartmouth
## 64.55450
## University of Miami
## 62.14600
## University of Michigan at Ann Arbor
## 68.95830
## University of Minnesota at Duluth
## 56.91581
## University of Minnesota at Morris
## 61.31256
## University of Minnesota Twin Cities
## 75.60106
## University of Mississippi
## 59.59977
## University of Missouri at Columbia
## 60.46382
## University of Missouri at Rolla
## 66.82527
## University of Missouri at Saint Louis
## 60.46382
## University of Mobile
## 50.11761
## University of Montevallo
## 54.09411
## University of Nebraska at Lincoln
## 80.39210
## University of New England
## 58.72042
## University of New Hampshire
## 61.31256
## University of North Carolina at Asheville
## 56.91581
## University of North Carolina at Chapel Hill
## 66.82527
## University of North Carolina at Charlotte
## 53.12294
## University of North Carolina at Greensboro
## 62.14600
## University of North Carolina at Wilmington
## 60.46382
## University of North Dakota
## 61.31256
## University of North Florida
## 59.59977
## University of North Texas
## 52.13647
## University of Northern Colorado
## 54.09411
## University of Northern Iowa
## 68.95830
## University of Notre Dame
## 79.65873
## University of Oklahoma
## 56.91581
## University of Oregon
## 58.72042
## University of Pennsylvania
## 76.11313
## University of Pittsburgh-Main Campus
## 55.99055
## University of Portland
## 62.14600
## University of Puget Sound
## 62.14600
## University of Rhode Island
## 53.12294
## University of Richmond
## 72.81118
## University of Rochester
## 66.82527
## University of San Diego
## 58.72042
## University of San Francisco
## 54.09411
## University of Sci. and Arts of Oklahoma
## 49.08522
## University of Scranton
## 77.55751
## University of South Carolina at Aiken
## 49.08522
## University of South Carolina at Columbia
## 62.96414
## University of South Florida
## 53.12294
## University of Southern California
## 55.99055
## University of Southern Colorado
## 45.89624
## University of Southern Indiana
## 65.32673
## University of Southern Mississippi
## 66.82527
## University of St. Thomas MN
## 58.72042
## University of St. Thomas TX
## 62.14600
## University of Tennessee at Knoxville
## 66.08365
## University of Texas at Arlington
## 50.11761
## University of Texas at Austin
## 56.91581
## University of Texas at San Antonio
## 49.08522
## University of the Arts
## 55.04998
## University of the Pacific
## 59.59977
## University of the South
## 81.67519
## University of Tulsa
## 55.99055
## University of Utah
## 55.04998
## University of Vermont
## 55.99055
## University of Virginia
## 66.08365
## University of Washington
## 55.99055
## University of West Florida
## 57.82577
## University of Wisconsin-Stout
## 62.14600
## University of Wisconsin-Superior
## 60.46382
## University of Wisconsin-Whitewater
## 61.31256
## University of Wisconsin at Green Bay
## 46.97454
## University of Wisconsin at Madison
## 64.55450
## University of Wisconsin at Milwaukee
## 54.09411
## University of Wyoming
## 58.72042
## Upper Iowa University
## 63.76697
## Ursinus College
## 77.09135
## Ursuline College
## 60.46382
## Valley City State University
## 68.26259
## Valparaiso University
## 66.82527
## Vanderbilt University
## 68.95830
## Vassar College
## 76.60989
## Villanova University
## 67.55158
## Virginia Commonwealth University
## 56.91581
## Virginia State University
## 56.91581
## Virginia Tech
## 64.55450
## Virginia Union University
## 54.09411
## Virginia Wesleyan College
## 59.59977
## Viterbo College
## 72.20729
## Voorhees College
## 49.08522
## Wabash College
## 82.47682
## Wagner College
## 66.82527
## Wake Forest University
## 75.60106
## Walsh University
## 73.39976
## Warren Wilson College
## 64.55450
## Wartburg College
## 75.60106
## Washington and Jefferson College
## 77.09135
## Washington and Lee University
## 79.26910
## Washington College
## 75.60106
## Washington State University
## 71.58810
## Washington University
## 72.20729
## Wayne State College
## 70.95361
## Waynesburg College
## 68.95830
## Webber College
## 50.11761
## Webster University
## 59.59977
## Wellesley College
## 81.37737
## Wells College
## 78.00836
## Wentworth Institute of Technology
## 54.09411
## Wesley College
## 60.46382
## Wesleyan University
## 76.60989
## West Chester University of Penn.
## 61.31256
## West Liberty State College
## 55.99055
## West Virginia Wesleyan College
## 78.00836
## Western Carolina University
## 55.04998
## Western Maryland College
## 76.60989
## Western Michigan University
## 56.91581
## Western New England College
## 60.46382
## Western State College of Colorado
## 50.11761
## Western Washington University
## 55.99055
## Westfield State College
## 64.55450
## Westminster College MO
## 64.55450
## Westminster College
## 77.55751
## Westminster College of Salt Lake City
## 73.97304
## Westmont College
## 62.14600
## Wheaton College IL
## 77.09135
## Westminster College PA
## 77.55751
## Wheeling Jesuit College
## 69.63871
## Whitman College
## 81.37737
## Whittier College
## 70.95361
## Whitworth College
## 64.55450
## Widener University
## 63.76697
## Wilkes University
## 67.55158
## Willamette University
## 75.60106
## William Jewell College
## 63.76697
## William Woods University
## 61.31256
## Williams College
## 84.05532
## Wilson College
## 78.44391
## Wingate College
## 54.09411
## Winona State University
## 62.96414
## Winthrop University
## 68.95830
## Wisconsin Lutheran College
## 68.95830
## Wittenberg University
## 70.95361
## Wofford College
## 78.00836
## Worcester Polytechnic Institute
## 73.97304
## Worcester State College
## 59.59977
## Xavier University
## 72.20729
## Xavier University of Louisiana
## 64.55450
## Yale University
## 80.73583
## York College of Pennsylvania
## 70.30381model.matrix(ajuste2) ### Matriz do modelo (matriz X)## (Intercept) perc.alumni
## Abilene Christian University 1 12
## Adelphi University 1 16
## Adrian College 1 30
## Agnes Scott College 1 37
## Alaska Pacific University 1 2
## Albertson College 1 11
## Albertus Magnus College 1 26
## Albion College 1 37
## Albright College 1 23
## Alderson-Broaddus College 1 15
## Alfred University 1 31
## Allegheny College 1 41
## Allentown Coll. of St. Francis de Sales 1 21
## Alma College 1 32
## Alverno College 1 26
## American International College 1 19
## Amherst College 1 63
## Anderson University 1 14
## Andrews University 1 18
## Angelo State University 1 5
## Antioch University 1 35
## Appalachian State University 1 14
## Aquinas College 1 25
## Arizona State University Main campus 1 5
## Arkansas College (Lyon College) 1 24
## Arkansas Tech University 1 5
## Assumption College 1 30
## Auburn University-Main Campus 1 18
## Augsburg College 1 31
## Augustana College IL 1 40
## Augustana College 1 30
## Austin College 1 33
## Averett College 1 11
## Baker University 1 21
## Baldwin-Wallace College 1 20
## Barat College 1 35
## Bard College 1 30
## Barnard College 1 33
## Barry University 1 11
## Baylor University 1 38
## Beaver College 1 30
## Bellarmine College 1 31
## Belmont Abbey College 1 10
## Belmont University 1 19
## Beloit College 1 26
## Bemidji State University 1 16
## Benedictine College 1 18
## Bennington College 1 33
## Bentley College 1 20
## Berry College 1 17
## Bethany College 1 29
## Bethel College KS 1 32
## Bethel College 1 13
## Bethune Cookman College 1 9
## Birmingham-Southern College 1 34
## Blackburn College 1 53
## Bloomsburg Univ. of Pennsylvania 1 19
## Bluefield College 1 3
## Bluffton College 1 19
## Boston University 1 16
## Bowdoin College 1 52
## Bowling Green State University 1 14
## Bradford College 1 21
## Bradley University 1 21
## Brandeis University 1 24
## Brenau University 1 12
## Brewton-Parker College 1 10
## Briar Cliff College 1 26
## Bridgewater College 1 24
## Brigham Young University at Provo 1 40
## Brown University 1 39
## Bryn Mawr College 1 49
## Bucknell University 1 36
## Buena Vista College 1 10
## Butler University 1 29
## Cabrini College 1 36
## Caldwell College 1 25
## California Lutheran University 1 17
## California Polytechnic-San Luis 1 13
## California State University at Fresno 1 8
## Calvin College 1 41
## Campbell University 1 34
## Campbellsville College 1 13
## Canisius College 1 26
## Capital University 1 27
## Capitol College 1 24
## Carleton College 1 60
## Carnegie Mellon University 1 31
## Carroll College 1 25
## Carson-Newman College 1 16
## Carthage College 1 22
## Case Western Reserve University 1 29
## Castleton State College 1 8
## Catawba College 1 27
## Catholic University of America 1 18
## Cazenovia College 1 20
## Cedar Crest College 1 39
## Cedarville College 1 34
## Centenary College 1 20
## Centenary College of Louisiana 1 25
## Center for Creative Studies 1 4
## Central College 1 29
## Central Connecticut State University 1 4
## Central Missouri State University 1 4
## Central Washington University 1 0
## Central Wesleyan College 1 18
## Centre College 1 60
## Chapman University 1 6
## Chatham College 1 37
## Chestnut Hill College 1 29
## Christendom College 1 17
## Christian Brothers University 1 24
## Christopher Newport University 1 16
## Claflin College 1 31
## Claremont McKenna College 1 52
## Clark University 1 35
## Clarke College 1 27
## Clarkson University 1 32
## Clemson University 1 17
## Clinch Valley Coll. of the Univ. of Virginia 1 9
## Coe College 1 32
## Coker College 1 39
## Colby College 1 41
## Colgate University 1 45
## College Misericordia 1 23
## College of Charleston 1 18
## College of Mount St. Joseph 1 35
## College of Mount St. Vincent 1 35
## College of Notre Dame 1 7
## College of Notre Dame of Maryland 1 32
## College of Saint Benedict 1 26
## College of Saint Catherine 1 32
## College of Saint Elizabeth 1 23
## College of Saint Rose 1 28
## College of Santa Fe 1 7
## College of St. Joseph 1 19
## College of St. Scholastica 1 33
## College of the Holy Cross 1 55
## College of William and Mary 1 31
## College of Wooster 1 43
## Colorado College 1 51
## Colorado State University 1 10
## Columbia College MO 1 2
## Columbia College 1 34
## Columbia University 1 21
## Concordia College at St. Paul 1 18
## Concordia Lutheran College 1 9
## Concordia University CA 1 13
## Concordia University 1 13
## Connecticut College 1 40
## Converse College 1 31
## Cornell College 1 31
## Creighton University 1 32
## Culver-Stockton College 1 28
## Cumberland College 1 4
## D'Youville College 1 42
## Dana College 1 25
## Daniel Webster College 1 10
## Dartmouth College 1 49
## Davidson College 1 46
## Defiance College 1 19
## Delta State University 1 16
## Denison University 1 45
## DePauw University 1 31
## Dickinson College 1 39
## Dickinson State University 1 28
## Dillard University 1 12
## Doane College 1 42
## Dominican College of Blauvelt 1 5
## Dordt College 1 17
## Dowling College 1 7
## Drake University 1 24
## Drew University 1 28
## Drury College 1 35
## Duke University 1 44
## Earlham College 1 46
## East Carolina University 1 18
## East Tennessee State University 1 9
## East Texas Baptist University 1 7
## Eastern College 1 22
## Eastern Connecticut State University 1 14
## Eastern Illinois University 1 5
## Eastern Mennonite College 1 29
## Eastern Nazarene College 1 17
## Eckerd College 1 26
## Elizabethtown College 1 25
## Elmira College 1 21
## Elms College 1 21
## Elon College 1 34
## Embry Riddle Aeronautical University 1 4
## Emory & Henry College 1 51
## Emory University 1 28
## Emporia State University 1 4
## Erskine College 1 47
## Eureka College 1 31
## Evergreen State College 1 14
## Fairfield University 1 30
## Fayetteville State University 1 10
## Ferrum College 1 9
## Flagler College 1 9
## Florida Institute of Technology 1 7
## Florida International University 1 20
## Florida Southern College 1 10
## Florida State University 1 15
## Fontbonne College 1 24
## Fordham University 1 14
## Fort Lewis College 1 6
## Francis Marion University 1 8
## Franciscan University of Steubenville 1 8
## Franklin College 1 37
## Franklin Pierce College 1 16
## Freed-Hardeman University 1 13
## Fresno Pacific College 1 14
## Furman University 1 28
## Gannon University 1 18
## Gardner Webb University 1 12
## Geneva College 1 26
## George Fox College 1 22
## George Mason University 1 7
## George Washington University 1 15
## Georgetown College 1 28
## Georgetown University 1 27
## Georgia Institute of Technology 1 33
## Georgia State University 1 10
## Georgian Court College 1 27
## Gettysburg College 1 32
## Goldey Beacom College 1 4
## Gonzaga University 1 32
## Gordon College 1 32
## Goshen College 1 46
## Goucher College 1 34
## Grace College and Seminary 1 26
## Graceland College 1 24
## Grand Valley State University 1 9
## Green Mountain College 1 24
## Greensboro College 1 31
## Greenville College 1 16
## Grinnell College 1 54
## Grove City College 1 18
## Guilford College 1 30
## Gustavus Adolphus College 1 58
## Gwynedd Mercy College 1 22
## Hamilton College 1 60
## Hamline University 1 33
## Hampden - Sydney College 1 53
## Hampton University 1 9
## Hanover College 1 26
## Hardin-Simmons University 1 10
## Harding University 1 37
## Hartwick College 1 32
## Harvard University 1 52
## Harvey Mudd College 1 46
## Hastings College 1 17
## Hendrix College 1 26
## Hillsdale College 1 31
## Hiram College 1 34
## Hobart and William Smith Colleges 1 37
## Hofstra University 1 10
## Hollins College 1 48
## Hood College 1 34
## Hope College 1 40
## Houghton College 1 24
## Huntingdon College 1 9
## Huntington College 1 25
## Huron University 1 4
## Husson College 1 4
## Illinois Benedictine College 1 29
## Illinois College 1 30
## Illinois Institute of Technology 1 26
## Illinois State University 1 16
## Illinois Wesleyan University 1 34
## Immaculata College 1 33
## Incarnate Word College 1 21
## Indiana State University 1 8
## Indiana University at Bloomington 1 24
## Indiana Wesleyan University 1 15
## Iona College 1 14
## Iowa State University 1 22
## Ithaca College 1 25
## James Madison University 1 29
## Jamestown College 1 21
## Jersey City State College 1 10
## John Brown University 1 19
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## Xavier University of Louisiana 1 20
## Yale University 1 49
## York College of Pennsylvania 1 28
## I(perc.alumni^2)
## Abilene Christian University 144
## Adelphi University 256
## Adrian College 900
## Agnes Scott College 1369
## Alaska Pacific University 4
## Albertson College 121
## Albertus Magnus College 676
## Albion College 1369
## Albright College 529
## Alderson-Broaddus College 225
## Alfred University 961
## Allegheny College 1681
## Allentown Coll. of St. Francis de Sales 441
## Alma College 1024
## Alverno College 676
## American International College 361
## Amherst College 3969
## Anderson University 196
## Andrews University 324
## Angelo State University 25
## Antioch University 1225
## Appalachian State University 196
## Aquinas College 625
## Arizona State University Main campus 25
## Arkansas College (Lyon College) 576
## Arkansas Tech University 25
## Assumption College 900
## Auburn University-Main Campus 324
## Augsburg College 961
## Augustana College IL 1600
## Augustana College 900
## Austin College 1089
## Averett College 121
## Baker University 441
## Baldwin-Wallace College 400
## Barat College 1225
## Bard College 900
## Barnard College 1089
## Barry University 121
## Baylor University 1444
## Beaver College 900
## Bellarmine College 961
## Belmont Abbey College 100
## Belmont University 361
## Beloit College 676
## Bemidji State University 256
## Benedictine College 324
## Bennington College 1089
## Bentley College 400
## Berry College 289
## Bethany College 841
## Bethel College KS 1024
## Bethel College 169
## Bethune Cookman College 81
## Birmingham-Southern College 1156
## Blackburn College 2809
## Bloomsburg Univ. of Pennsylvania 361
## Bluefield College 9
## Bluffton College 361
## Boston University 256
## Bowdoin College 2704
## Bowling Green State University 196
## Bradford College 441
## Bradley University 441
## Brandeis University 576
## Brenau University 144
## Brewton-Parker College 100
## Briar Cliff College 676
## Bridgewater College 576
## Brigham Young University at Provo 1600
## Brown University 1521
## Bryn Mawr College 2401
## Bucknell University 1296
## Buena Vista College 100
## Butler University 841
## Cabrini College 1296
## Caldwell College 625
## California Lutheran University 289
## California Polytechnic-San Luis 169
## California State University at Fresno 64
## Calvin College 1681
## Campbell University 1156
## Campbellsville College 169
## Canisius College 676
## Capital University 729
## Capitol College 576
## Carleton College 3600
## Carnegie Mellon University 961
## Carroll College 625
## Carson-Newman College 256
## Carthage College 484
## Case Western Reserve University 841
## Castleton State College 64
## Catawba College 729
## Catholic University of America 324
## Cazenovia College 400
## Cedar Crest College 1521
## Cedarville College 1156
## Centenary College 400
## Centenary College of Louisiana 625
## Center for Creative Studies 16
## Central College 841
## Central Connecticut State University 16
## Central Missouri State University 16
## Central Washington University 0
## Central Wesleyan College 324
## Centre College 3600
## Chapman University 36
## Chatham College 1369
## Chestnut Hill College 841
## Christendom College 289
## Christian Brothers University 576
## Christopher Newport University 256
## Claflin College 961
## Claremont McKenna College 2704
## Clark University 1225
## Clarke College 729
## Clarkson University 1024
## Clemson University 289
## Clinch Valley Coll. of the Univ. of Virginia 81
## Coe College 1024
## Coker College 1521
## Colby College 1681
## Colgate University 2025
## College Misericordia 529
## College of Charleston 324
## College of Mount St. Joseph 1225
## College of Mount St. Vincent 1225
## College of Notre Dame 49
## College of Notre Dame of Maryland 1024
## College of Saint Benedict 676
## College of Saint Catherine 1024
## College of Saint Elizabeth 529
## College of Saint Rose 784
## College of Santa Fe 49
## College of St. Joseph 361
## College of St. Scholastica 1089
## College of the Holy Cross 3025
## College of William and Mary 961
## College of Wooster 1849
## Colorado College 2601
## Colorado State University 100
## Columbia College MO 4
## Columbia College 1156
## Columbia University 441
## Concordia College at St. Paul 324
## Concordia Lutheran College 81
## Concordia University CA 169
## Concordia University 169
## Connecticut College 1600
## Converse College 961
## Cornell College 961
## Creighton University 1024
## Culver-Stockton College 784
## Cumberland College 16
## D'Youville College 1764
## Dana College 625
## Daniel Webster College 100
## Dartmouth College 2401
## Davidson College 2116
## Defiance College 361
## Delta State University 256
## Denison University 2025
## DePauw University 961
## Dickinson College 1521
## Dickinson State University 784
## Dillard University 144
## Doane College 1764
## Dominican College of Blauvelt 25
## Dordt College 289
## Dowling College 49
## Drake University 576
## Drew University 784
## Drury College 1225
## Duke University 1936
## Earlham College 2116
## East Carolina University 324
## East Tennessee State University 81
## East Texas Baptist University 49
## Eastern College 484
## Eastern Connecticut State University 196
## Eastern Illinois University 25
## Eastern Mennonite College 841
## Eastern Nazarene College 289
## Eckerd College 676
## Elizabethtown College 625
## Elmira College 441
## Elms College 441
## Elon College 1156
## Embry Riddle Aeronautical University 16
## Emory & Henry College 2601
## Emory University 784
## Emporia State University 16
## Erskine College 2209
## Eureka College 961
## Evergreen State College 196
## Fairfield University 900
## Fayetteville State University 100
## Ferrum College 81
## Flagler College 81
## Florida Institute of Technology 49
## Florida International University 400
## Florida Southern College 100
## Florida State University 225
## Fontbonne College 576
## Fordham University 196
## Fort Lewis College 36
## Francis Marion University 64
## Franciscan University of Steubenville 64
## Franklin College 1369
## Franklin Pierce College 256
## Freed-Hardeman University 169
## Fresno Pacific College 196
## Furman University 784
## Gannon University 324
## Gardner Webb University 144
## Geneva College 676
## George Fox College 484
## George Mason University 49
## George Washington University 225
## Georgetown College 784
## Georgetown University 729
## Georgia Institute of Technology 1089
## Georgia State University 100
## Georgian Court College 729
## Gettysburg College 1024
## Goldey Beacom College 16
## Gonzaga University 1024
## Gordon College 1024
## Goshen College 2116
## Goucher College 1156
## Grace College and Seminary 676
## Graceland College 576
## Grand Valley State University 81
## Green Mountain College 576
## Greensboro College 961
## Greenville College 256
## Grinnell College 2916
## Grove City College 324
## Guilford College 900
## Gustavus Adolphus College 3364
## Gwynedd Mercy College 484
## Hamilton College 3600
## Hamline University 1089
## Hampden - Sydney College 2809
## Hampton University 81
## Hanover College 676
## Hardin-Simmons University 100
## Harding University 1369
## Hartwick College 1024
## Harvard University 2704
## Harvey Mudd College 2116
## Hastings College 289
## Hendrix College 676
## Hillsdale College 961
## Hiram College 1156
## Hobart and William Smith Colleges 1369
## Hofstra University 100
## Hollins College 2304
## Hood College 1156
## Hope College 1600
## Houghton College 576
## Huntingdon College 81
## Huntington College 625
## Huron University 16
## Husson College 16
## Illinois Benedictine College 841
## Illinois College 900
## Illinois Institute of Technology 676
## Illinois State University 256
## Illinois Wesleyan University 1156
## Immaculata College 1089
## Incarnate Word College 441
## Indiana State University 64
## Indiana University at Bloomington 576
## Indiana Wesleyan University 225
## Iona College 196
## Iowa State University 484
## Ithaca College 625
## James Madison University 841
## Jamestown College 441
## Jersey City State College 100
## John Brown University 361
## John Carroll University 784
## Johns Hopkins University 1444
## Johnson State College 225
## Judson College 900
## Juniata College 1369
## Kansas State University 484
## Kansas Wesleyan University 196
## Keene State College 169
## Kentucky Wesleyan College 1024
## Kenyon College 2116
## Keuka College 1849
## King's College 1369
## King College 625
## Knox College 1089
## La Roche College 196
## La Salle University 81
## Lafayette College 1444
## LaGrange College 144
## Lake Forest College 361
## Lakeland College 625
## Lamar University 144
## Lambuth University 100
## Lander University 121
## Lawrence University 3249
## Le Moyne College 784
## Lebanon Valley College 900
## Lehigh University 1849
## Lenoir-Rhyne College 400
## Lesley College 324
## LeTourneau University 529
## Lewis and Clark College 441
## Lewis University 100
## Lincoln Memorial University 1225
## Lincoln University 64
## Lindenwood College 81
## Linfield College 1156
## Livingstone College 256
## Lock Haven University of Pennsylvania 196
## Longwood College 529
## Loras College 576
## Louisiana College 121
## Louisiana State University at Baton Rouge 121
## Louisiana Tech University 169
## Loyola College 729
## Loyola Marymount University 100
## Loyola University 196
## Loyola University Chicago 225
## Luther College 1444
## Lycoming College 1024
## Lynchburg College 576
## Lyndon State College 225
## Macalester College 1369
## MacMurray College 1089
## Malone College 256
## Manchester College 400
## Manhattan College 625
## Manhattanville College 576
## Mankato State University 121
## Marian College of Fond du Lac 441
## Marietta College 900
## Marist College 1156
## Marquette University 625
## Marshall University 100
## Mary Baldwin College 2500
## Mary Washington College 900
## Marymount College Tarrytown 900
## Marymount Manhattan College 400
## Marymount University 289
## Maryville College 1849
## Maryville University 169
## Marywood College 900
## Massachusetts Institute of Technology 1225
## Mayville State University 121
## McKendree College 441
## McMurry University 121
## McPherson College 2025
## Mercer University 225
## Mercyhurst College 841
## Meredith College 1089
## Merrimack College 484
## Mesa State College 144
## Messiah College 900
## Miami University at Oxford 400
## Michigan State University 81
## Michigan Technological University 625
## MidAmerica Nazarene College 400
## Millersville University of Penn. 400
## Milligan College 256
## Millikin University 625
## Millsaps College 1444
## Milwaukee School of Engineering 529
## Mississippi College 324
## Mississippi State University 400
## Mississippi University for Women 64
## Missouri Southern State College 81
## Missouri Valley College 256
## Monmouth College IL 1849
## Monmouth College 225
## Montana College of Mineral Sci. & Tech. 961
## Montana State University 64
## Montclair State University 81
## Montreat-Anderson College 16
## Moorhead State University 361
## Moravian College 784
## Morehouse College 100
## Morningside College 1024
## Morris College 1156
## Mount Holyoke College 2601
## Mount Marty College 1444
## Mount Mary College 676
## Mount Mercy College 900
## Mount Saint Clare College 1849
## Mount Saint Mary's College 1296
## Mount Saint Mary College 529
## Mount St. Mary's College 361
## Mount Union College 1225
## Mount Vernon Nazarene College 49
## Muhlenberg College 1521
## Murray State University 729
## Muskingum College 729
## National-Louis University 4
## Nazareth College of Rochester 576
## New Jersey Institute of Technology 361
## New Mexico Institute of Mining and Tech. 121
## New York University 256
## Newberry College 1024
## Niagara University 400
## North Adams State College 289
## North Carolina A. & T. State University 81
## North Carolina State University at Raleigh 441
## North Carolina Wesleyan College 121
## North Central College 1089
## North Dakota State University 576
## North Park College 576
## Northeast Missouri State University 169
## Northeastern University 289
## Northern Arizona University 49
## Northern Illinois University 121
## Northwest Missouri State University 529
## Northwest Nazarene College 400
## Northwestern College 1156
## Northwestern University 625
## Norwich University 484
## Notre Dame College 676
## Oakland University 169
## Oberlin College 2209
## Occidental College 900
## Oglethorpe University 729
## Ohio Northern University 961
## Ohio University 169
## Ohio Wesleyan University 1024
## Oklahoma Baptist University 324
## Oklahoma Christian University 64
## Oklahoma State University 196
## Otterbein College 900
## Ouachita Baptist University 100
## Our Lady of the Lake University 625
## Pace University 100
## Pacific Lutheran University 529
## Pacific Union College 144
## Pacific University 484
## Pembroke State University 25
## Pennsylvania State Univ. Main Campus 361
## Pepperdine University 169
## Peru State College 576
## Pfeiffer College 169
## Philadelphia Coll. of Textiles and Sci. 225
## Phillips University 361
## Piedmont College 289
## Pikeville College 196
## Pitzer College 121
## Point Loma Nazarene College 361
## Point Park College 100
## Polytechnic University 196
## Prairie View A. and M. University 1
## Presbyterian College 1764
## Princeton University 2916
## Providence College 1225
## Purdue University at West Lafayette 225
## Queens College 1296
## Quincy University 1024
## Quinnipiac College 1089
## Radford University 81
## Ramapo College of New Jersey 64
## Randolph-Macon College 1444
## Randolph-Macon Woman's College 576
## Reed College 1369
## Regis College 1369
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## Rockhurst College 441
## Rocky Mountain College 729
## Roger Williams University 64
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## Rowan College of New Jersey 36
## Rutgers at New Brunswick 361
## Rutgers State University at Camden 144
## Rutgers State University at Newark 256
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## Saint Cloud State University 100
## Saint Francis College IN 16
## Saint Francis College 576
## Saint John's University 1444
## Saint Joseph's College IN 361
## Saint Joseph's College 64
## Saint Joseph's University 169
## Saint Joseph College 1024
## Saint Louis University 361
## Saint Mary's College 961
## Saint Mary's College of Minnesota 361
## Saint Mary-of-the-Woods College 1369
## Saint Michael's College 1156
## Saint Olaf College 961
## Saint Peter's College 484
## Saint Vincent College 961
## Saint Xavier University 225
## Salem-Teikyo University 81
## Salem College 2116
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## San Diego State University 49
## Santa Clara University 361
## Sarah Lawrence College 324
## Savannah Coll. of Art and Design 676
## Schreiner College 529
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## Seattle Pacific University 400
## Seattle University 256
## Seton Hall University 225
## Seton Hill College 1369
## Shippensburg University of Penn. 169
## Shorter College 324
## Siena College 1764
## Siena Heights College 289
## Simmons College 1089
## Simpson College 1296
## Sioux Falls College 49
## Skidmore College 841
## Smith College 1936
## South Dakota State University 841
## Southeast Missouri State University 64
## Southeastern Oklahoma State Univ. 81
## Southern California College 121
## Southern Illinois University at Edwardsville 64
## Southern Methodist University 289
## Southwest Baptist University 169
## Southwest Missouri State University 121
## Southwest State University 961
## Southwestern Adventist College 144
## Southwestern College 144
## Southwestern University 1225
## Spalding University 1600
## Spelman College 324
## Spring Arbor College 81
## St. Bonaventure University 1024
## St. John's College 676
## St. John Fisher College 841
## St. Lawrence University 1444
## St. Martin's College 64
## St. Mary's College of California 289
## St. Mary's College of Maryland 529
## St. Mary's University of San Antonio 49
## St. Norbert College 1296
## St. Paul's College 64
## St. Thomas Aquinas College 169
## Stephens College 289
## Stetson University 576
## Stevens Institute of Technology 1089
## Stockton College of New Jersey 49
## Stonehill College 900
## SUNY at Albany 256
## SUNY at Binghamton 225
## SUNY at Buffalo 225
## SUNY at Stony Brook 49
## SUNY College at Brockport 196
## SUNY College at Oswego 441
## SUNY College at Buffalo 144
## SUNY College at Cortland 289
## SUNY College at Fredonia 100
## SUNY College at Geneseo 625
## SUNY College at New Paltz 64
## SUNY College at Plattsburgh 256
## SUNY College at Potsdam 289
## SUNY College at Purchase 64
## Susquehanna University 1369
## Sweet Briar College 2304
## Syracuse University 169
## Tabor College 225
## Talladega College 49
## Taylor University 1024
## Tennessee Wesleyan College 256
## Texas A&M Univ. at College Station 841
## Texas A&M University at Galveston 256
## Texas Christian University 529
## Texas Lutheran College 576
## Texas Southern University 441
## Texas Wesleyan University 100
## The Citadel 289
## Thiel College 256
## Tiffin University 1600
## Transylvania University 1681
## Trenton State College 36
## Tri-State University 576
## Trinity College CT 2304
## Trinity College DC 1369
## Trinity College VT 676
## Trinity University 400
## Tulane University 441
## Tusculum College 784
## Tuskegee University 49
## Union College KY 81
## Union College NY 2401
## Univ. of Wisconsin at OshKosh 196
## University of Alabama at Birmingham 256
## University of Arkansas at Fayetteville 100
## University of California at Berkeley 100
## University of California at Irvine 121
## University of Central Florida 81
## University of Charleston 100
## University of Chicago 1296
## University of Cincinnati 36
## University of Connecticut at Storrs 256
## University of Dallas 676
## University of Dayton 625
## University of Delaware 225
## University of Denver 441
## University of Detroit Mercy 196
## University of Dubuque 324
## University of Evansville 676
## University of Florida 400
## University of Georgia 484
## University of Hartford 81
## University of Hawaii at Manoa 36
## University of Illinois - Urbana 169
## University of Illinois at Chicago 36
## University of Indianapolis 529
## University of Kansas 289
## University of La Verne 529
## University of Louisville 576
## University of Maine at Farmington 676
## University of Maine at Machias 16
## University of Maine at Presque Isle 121
## University of Maryland at Baltimore County 36
## University of Maryland at College Park 144
## University of Massachusetts at Amherst 225
## University of Massachusetts at Dartmouth 400
## University of Miami 289
## University of Michigan at Ann Arbor 676
## University of Minnesota at Duluth 121
## University of Minnesota at Morris 256
## University of Minnesota Twin Cities 1369
## University of Mississippi 196
## University of Missouri at Columbia 225
## University of Missouri at Rolla 529
## University of Missouri at Saint Louis 225
## University of Mobile 16
## University of Montevallo 64
## University of Nebraska at Lincoln 2304
## University of New England 169
## University of New Hampshire 256
## University of North Carolina at Asheville 121
## University of North Carolina at Chapel Hill 529
## University of North Carolina at Charlotte 49
## University of North Carolina at Greensboro 289
## University of North Carolina at Wilmington 225
## University of North Dakota 256
## University of North Florida 196
## University of North Texas 36
## University of Northern Colorado 64
## University of Northern Iowa 676
## University of Notre Dame 2116
## University of Oklahoma 121
## University of Oregon 169
## University of Pennsylvania 1444
## University of Pittsburgh-Main Campus 100
## University of Portland 289
## University of Puget Sound 289
## University of Rhode Island 49
## University of Richmond 1024
## University of Rochester 529
## University of San Diego 169
## University of San Francisco 64
## University of Sci. and Arts of Oklahoma 9
## University of Scranton 1681
## University of South Carolina at Aiken 9
## University of South Carolina at Columbia 324
## University of South Florida 49
## University of Southern California 100
## University of Southern Colorado 0
## University of Southern Indiana 441
## University of Southern Mississippi 529
## University of St. Thomas MN 169
## University of St. Thomas TX 289
## University of Tennessee at Knoxville 484
## University of Texas at Arlington 16
## University of Texas at Austin 121
## University of Texas at San Antonio 9
## University of the Arts 81
## University of the Pacific 196
## University of the South 2704
## University of Tulsa 100
## University of Utah 81
## University of Vermont 100
## University of Virginia 484
## University of Washington 100
## University of West Florida 144
## University of Wisconsin-Stout 289
## University of Wisconsin-Superior 225
## University of Wisconsin-Whitewater 256
## University of Wisconsin at Green Bay 1
## University of Wisconsin at Madison 400
## University of Wisconsin at Milwaukee 64
## University of Wyoming 169
## Upper Iowa University 361
## Ursinus College 1600
## Ursuline College 225
## Valley City State University 625
## Valparaiso University 529
## Vanderbilt University 676
## Vassar College 1521
## Villanova University 576
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## Virginia Tech 400
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## Virginia Wesleyan College 196
## Viterbo College 961
## Voorhees College 9
## Wabash College 3025
## Wagner College 529
## Wake Forest University 1369
## Walsh University 1089
## Warren Wilson College 400
## Wartburg College 1369
## Washington and Jefferson College 1600
## Washington and Lee University 2025
## Washington College 1369
## Washington State University 900
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## Wayne State College 841
## Waynesburg College 676
## Webber College 16
## Webster University 196
## Wellesley College 2601
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## Wentworth Institute of Technology 64
## Wesley College 225
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## West Chester University of Penn. 256
## West Liberty State College 100
## West Virginia Wesleyan College 1764
## Western Carolina University 81
## Western Maryland College 1521
## Western Michigan University 121
## Western New England College 225
## Western State College of Colorado 16
## Western Washington University 100
## Westfield State College 400
## Westminster College MO 400
## Westminster College 1681
## Westminster College of Salt Lake City 1156
## Westmont College 289
## Wheaton College IL 1600
## Westminster College PA 1681
## Wheeling Jesuit College 729
## Whitman College 2601
## Whittier College 841
## Whitworth College 400
## Widener University 361
## Wilkes University 576
## Willamette University 1369
## William Jewell College 361
## William Woods University 256
## Williams College 4096
## Wilson College 1849
## Wingate College 64
## Winona State University 324
## Winthrop University 676
## Wisconsin Lutheran College 676
## Wittenberg University 841
## Wofford College 1764
## Worcester Polytechnic Institute 1156
## Worcester State College 196
## Xavier University 961
## Xavier University of Louisiana 400
## Yale University 2401
## York College of Pennsylvania 784
## attr(,"assign")
## [1] 0 1 2vcov(ajuste2) ### Matriz de variâncias e covariâncias dos estimadores.## (Intercept) perc.alumni I(perc.alumni^2)
## (Intercept) 3.526090777 -0.2679031981 4.254628e-03
## perc.alumni -0.267903198 0.0242157266 -4.217689e-04
## I(perc.alumni^2) 0.004254628 -0.0004217689 7.959571e-06### Vamos visualizar o ajuste do modelo
ggplot(College, aes(x = perc.alumni, y = Grad.Rate)) + geom_point() +
stat_smooth(method = "lm", formula = y ~ x + I(x^2)) +
theme_bw(base_size = 14)
### Extraindo os intervalos de confiança (95%) para os parâmetros
confint(ajuste2)## 2.5 % 97.5 %
## (Intercept) 42.2100816 49.582406847
## perc.alumni 0.7804723 1.391423440
## I(perc.alumni^2) -0.0131900 -0.002113503### vamos realizar algumas predições. Considere faculdades com os seguintes
### percentuais de ex-alunos contribuintes: 13, 28 e 45
new_data <- data.frame(perc.alumni = c(13,28,45))
predict(ajuste2, newdata = new_data, se.fit = TRUE)## $fit
## 1 2 3
## 58.72042 70.30381 79.26910
##
## $se.fit
## 1 2 3
## 0.6820636 0.7449543 1.2061247
##
## $df
## [1] 774
##
## $residual.scale
## [1] 14.91414### Estimativas pontuais e erros padrões (nota: erros padrões para a resposta
### média)
predict(ajuste2, newdata = new_data, interval = 'confidence') ## fit lwr upr
## 1 58.72042 57.38151 60.05933
## 2 70.30381 68.84144 71.76618
## 3 79.26910 76.90143 81.63676### Estimativas pontiais e intervalos de confiança (95%) para a resposta média.
predict(ajuste2, newdata = new_data, interval = 'prediction') ## fit lwr upr
## 1 58.72042 29.41286 88.02798
## 2 70.30381 40.99035 99.61727
## 3 79.26910 49.89656 108.64164### Estimativas pontiais e intervalos de confiança (95%) para a predição
### de uma nova observação.### Vamos plotar as bandas de confiança e predição. Para isso, vamos preparar
### uma base de dados com os valores ajustados e os ICs(95%) para a resposta
### média e para as predições.
pred_int <- predict(ajuste2, interval="prediction")
med_int <- predict(ajuste2, interval="confidence")
data_pred <- data.frame(pred_lwr = pred_int[,'lwr'], pred_upr = pred_int[,'upr'],
med_lwr = med_int[,'lwr'], med_upr = med_int[,'upr'],
fit = med_int[,'fit'], perc.alumni = College$perc.alumni,
Grad.Rate = College$Grad.Rate)
ggplot(data_pred, aes(x = perc.alumni, y = Grad.Rate))+
geom_point() +
geom_line(aes(y=med_lwr), color = "red", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=med_upr), color = "red", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=pred_lwr), color = "green", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=pred_upr), color = "green", linetype = "dashed", linewidth = 1.25) +
geom_line(aes(y=fit), color = "black", linetype = "dashed", linewidth = 1.25) +
theme_bw(base_size = 14)### Diagnóstico do ajuste (análise de resíduos)
### Vamos produzir alguns gráficos para os resíduos
fit_aj2 <- fitted(ajuste2) ### Vetor de valores ajustados
resid_aj2 <- rstandard(ajuste2) ### Vetor de resíduos padronizados
data_fit <- data.frame(y = College$Grad.Rate, fit_aj2, resid_aj2)
ggplot(data_fit, aes(x=y, y=fit_aj2)) + geom_point() + stat_smooth(method="lm") +
theme_bw(base_size = 14)
### Gráfico de valores observados versus valores ajustados.
ggplot(ajuste2, aes(x=fit_aj2, y=resid_aj2)) + geom_point() +
stat_smooth(method="loess") + geom_hline(yintercept=0, col="red", linetype="dashed") +
theme_bw(base_size = 14) +
xlab('Valores ajustados') +
ylab('Resíduos')
### Gráfico de resíduos versus valores ajustados
qqPlot(ajuste2)
### Gráfico quantil-quantil para os resíduos### Parte 2 - Regressão linear múltipla, incluindo todas as variáveis da base como explicativas (exceto a taxa de formação, que é a resposta)
ajuste_p2 <- lm(Grad.Rate ~., data = College)
### Ajuste da regressão linear múltipla. A especificação "~." indica que
### todas as demais variáveis da base devem ser incluídas como explicativas.
### A título de ilustração, se quiséssemos ajustar um modelo apenas com as
### variáveis "Private", "Apps" e "Accept":
summary(ajuste_p2) ##
## Call:
## lm(formula = Grad.Rate ~ ., data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.897 -7.132 -0.292 7.213 54.056
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.8736716 4.8480858 6.987 6.15e-12 ***
## PrivateYes 3.3813758 1.6965147 1.993 0.046605 *
## Apps 0.0012984 0.0004418 2.939 0.003390 **
## Accept -0.0006961 0.0008627 -0.807 0.419995
## Enroll 0.0021593 0.0023081 0.936 0.349814
## Top10perc 0.0548964 0.0717587 0.765 0.444501
## Top25perc 0.1351288 0.0549667 2.458 0.014179 *
## F.Undergrad -0.0004712 0.0004008 -1.176 0.240138
## P.Undergrad -0.0014836 0.0003902 -3.802 0.000155 ***
## Outstate 0.0010174 0.0002334 4.359 1.49e-05 ***
## Room.Board 0.0019143 0.0005908 3.240 0.001246 **
## Books -0.0022205 0.0029168 -0.761 0.446739
## Personal -0.0016635 0.0007698 -2.161 0.031000 *
## PhD 0.0872827 0.0568102 1.536 0.124859
## Terminal -0.0747023 0.0623172 -1.199 0.231002
## S.F.Ratio 0.0758222 0.1593102 0.476 0.634254
## perc.alumni 0.2793343 0.0491750 5.680 1.91e-08 ***
## Expend -0.0004565 0.0001542 -2.961 0.003163 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.75 on 759 degrees of freedom
## Multiple R-squared: 0.4615, Adjusted R-squared: 0.4495
## F-statistic: 38.27 on 17 and 759 DF, p-value: < 2.2e-16ajuste_p2 <- lm(Grad.Rate ~.-Top25perc, data = College)
## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"
summary(ajuste_p2) ##
## Call:
## lm(formula = Grad.Rate ~ . - Top25perc, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.786 -7.120 -0.257 7.171 54.050
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.3495104 4.7580380 7.640 6.57e-14 ***
## PrivateYes 3.3824039 1.7021347 1.987 0.047264 *
## Apps 0.0011831 0.0004407 2.685 0.007421 **
## Accept -0.0003936 0.0008568 -0.459 0.646059
## Enroll 0.0015802 0.0023036 0.686 0.492938
## Top10perc 0.1951766 0.0436553 4.471 8.98e-06 ***
## F.Undergrad -0.0003934 0.0004009 -0.981 0.326837
## P.Undergrad -0.0014830 0.0003915 -3.788 0.000164 ***
## Outstate 0.0010161 0.0002342 4.339 1.63e-05 ***
## Room.Board 0.0019288 0.0005927 3.254 0.001188 **
## Books -0.0020595 0.0029258 -0.704 0.481692
## Personal -0.0016865 0.0007723 -2.184 0.029273 *
## PhD 0.0876416 0.0569982 1.538 0.124558
## Terminal -0.0565580 0.0620836 -0.911 0.362585
## S.F.Ratio 0.0725205 0.1598322 0.454 0.650153
## perc.alumni 0.2896423 0.0491582 5.892 5.73e-09 ***
## Expend -0.0005259 0.0001521 -3.459 0.000573 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.79 on 760 degrees of freedom
## Multiple R-squared: 0.4573, Adjusted R-squared: 0.4458
## F-statistic: 40.02 on 16 and 760 DF, p-value: < 2.2e-16Exemplo de numero de comodos e area em preço do imovel. Em sepada os dois são significantes. No modelo tudo junto ajutado a area é positiva e os comodos são negativos (mais comodos numa mesma area, ou seja comodos menores, menos preço).
ajuste_p2 <- lm(Grad.Rate ~ Expend, data = College)
## Todas menos a top 25. pq o top 10 e top 25 "se sobrepoe"
## "ajustado os efeitos das outras variaveis"
summary(ajuste_p2) ##
## Call:
## lm(formula = Grad.Rate ~ Expend, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -60.179 -10.275 0.238 10.377 55.058
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.306e+01 1.194e+00 44.42 <2e-16 ***
## Expend 1.284e-03 1.088e-04 11.80 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.83 on 775 degrees of freedom
## Multiple R-squared: 0.1524, Adjusted R-squared: 0.1513
## F-statistic: 139.3 on 1 and 775 DF, p-value: < 2.2e-16ajuste_p2_ilustrativo <- lm(Grad.Rate ~ Private + Apps + Accept, data = College)
summary(ajuste_p2_ilustrativo) ##
## Call:
## lm(formula = Grad.Rate ~ Private + Apps + Accept, data = College)
##
## Residuals:
## Min 1Q Median 3Q Max
## -51.513 -8.856 0.228 10.516 50.053
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 48.5068491 1.4308933 33.900 < 2e-16 ***
## PrivateYes 17.7697914 1.3831646 12.847 < 2e-16 ***
## Apps 0.0030589 0.0004228 7.235 1.12e-12 ***
## Accept -0.0025493 0.0006842 -3.726 0.000209 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.09 on 773 degrees of freedom
## Multiple R-squared: 0.2316, Adjusted R-squared: 0.2287
## F-statistic: 77.68 on 3 and 773 DF, p-value: < 2.2e-16### Resumo do ajuste. Observe as variáveis com efeito significativo e os respectivos sinais.
### Variáveis com p-valor (Pr(>|t|)) < 0.05 podem ser consideradas com efeito
### significativo na taxa de formados. Desta forma, as variáveis com efeito
### significativo na taxa de formados são Private (Yes), Apps, Top25perc, Outstate,
### Room.Board, 0.2793343; Já as variáveis com efeito negativo na taxa de formados
### são P.Undergrad, Personal e Expend.### Apenas para fins de discussão, vamos ajustar um modelo de regressão tendo como
### única variável explicativa o gasto por aluno.
ajuste_temp <- lm(Grad.Rate ~ Expend, data = College)
summary(ajuste_temp)
### Compare o efeito do gasto por aluno na taxa de formação produzida pelo
### modelo em que ajustamos também os efeitos das demais variáveis com o
### efeito produzido pelo modelo em que as demais variáveis não são consideradas.
### Qual a diferença? Como você a justifica?par(mfrow = c(2,2))
plot(ajuste_p2, which = 1:4)
### Gráficos de resíduos.
### O gráfico do canto superior esquerdo indica que a variância dos resíduos varia
### um pouco com a média (fitted values), e que os resíduos apresentam algum desvio
### da normalidade (conforme o qqplot). Ainda, conforme o gráfico do canto inferior
### direito, algumas observações podem ser identificadas como possivelmente influentes,
### produzindo maiores valores para a distância de Cook.
### Embora tenhamos alguns indicativos (ainda que não tão severos) de falta de ajuste
### da regressão linear, para fins didáticos vamos seguir a análise com esse tipo
### de modelagem.### Agora, vamos para a etapa de seleção de covariáveis. Para isso, vamos
### usar os recursos do pacote leaps. Vamos consultar a documentação da função
### regsubsets.
help("regsubsets")
all_reg <- regsubsets(Grad.Rate ~ ., method = "exhaustive", nvmax = 18, data = College)
### Explorando TODAS as regressões possíveis
all_regplot(all_reg, scale="r2") ### Resultados baseados no R2
plot(all_reg, scale="adjr2") ### Resultados baseados no R2 ajustado
plot(all_reg, scale="bic") ### Resultados baseados no BIC
### Nesses gráficos avaliamos os resultados dos critérios para os melhores
### modelos ajustados com cada número de covariáveis.
s1 <- summary(all_reg, matrix.logical=TRUE)
s1 ### A matriz lógica permite identificar as covariáveis selecionadas
### em cada modelo.
s1$rsq ### Valores de R2 para cada um dos modelos selecionados
s1$adjr2 ### Valores de R2 ajustado para cada um dos modelos selecionados
s1$bic ### Valores de BIC para cada um dos modelos selecionados
which.max(s1$adjr2)
coef(all_reg, id = 12)
### O modelo com doze covariáveis produziu maior valor de R2 ajustado.
which.min(s1$bic)
coef(all_reg, id = 7)
### O modelo com sete covariáveis produziu menor valor de BIC.
which.max(s1$rsq)
coef(all_reg, id = 17)
### O modelo com 17 covariáveis produziu menor valor de R2 (obviamente).
### Vamos produzir alguns gráficos usando os resultados desta análise.
n_cov <- 1:17
plot(n_cov, s1$bic, type = 'b', xlab = 'Número de covariáveis',
ylab = 'BIC', las = 1, pch = 20)
axis(1,1:17)
plot(n_cov, s1$adjr2, type = 'b', xlab = 'Número de covariáveis',
ylab = 'Adjusted R2', las = 1, pch = 20)
axis(1,1:17)
plot(n_cov, s1$rsq, type = 'b', xlab = 'Número de covariáveis',
ylab = 'R2', las = 1, pch = 20)
axis(1,1:17)
### Para finalizar, vamos aplicar os algoritmos de seleção do tipo
### stepwise. Primeiramente fixando k = 2, estamos definindo o AIC como critério de
### seleção, temos:
### Método backward
step_back_AIC <- step(aj_full, direction = "backward", data = College, k = 2)
summary(step_back_AIC)
### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)
aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)
step_for_AIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper),
data = College, k = 2)
summary(step_for_AIC)
### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_AIC <- step(aj_full, direction = "both", data = College, k = 2)
summary(step_both_AIC)
### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_AIC, step_for_AIC, step_both_AIC))
### Agora fixando k = log(n) (critério BIC):
### Método backward
step_back_BIC <- step(aj_full, direction = "backward", data = College, k = log(nrow(College)))
summary(step_back_BIC)
### Método forward. Para o método forward devemos definir o escopo da seleção
### (menor e maior modelo). O menor seria o modelo nulo (apenas com o intercepto),
### enquanto o maior seria o modelo com todas as covariáveis.
aj_lower <- lm(Grad.Rate~1, data = College)
aj_upper <- lm(Grad.Rate~., data = College)
formula(aj_upper)
step_for_BIC <- step(aj_lower, direction = "forward", scope=formula(aj_upper),
data = College, k = log(nrow(College)))
summary(step_for_BIC)
### Finalmente, o algoritmo que considera tanto exclusão quanto inclusão de
### covariáveis a cada passo
step_both_BIC <- step(aj_full, direction = "both", data = College, k = log(nrow(College)))
summary(step_both_BIC)
### Vamos comparar os ajustes.
data.frame(compareCoefs(step_back_BIC, step_for_BIC, step_both_BIC))